{"as_of":"2026-08-16T23:18:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:e11c6d6e4aff51bf8e3a133073d7db634da175933db5d72990713749466448e8","coverage":[{"denominator":23,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":23,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-14T14:14:48.829265Z","state":"measured"},{"denominator":23,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":23,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-16T06:30:59.297886+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/1908.03616/citation-record","integrity":"/paper/1908.03616/integrity","json":"/paper/1908.03616/citation-record.json","paper":"/paper/1908.03616"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.203378Z","title":"4 Eisenstein series In preparation for the proof of this section’ s main theorem, we determine the space of cusp expan- sions of Eisenstein series","venue":null,"work_id":"b6b8431c-be27-448d-9b64-5117d8f85ed0","year":null},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.722631Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:95fe46037d48c6c04b983dc242d39e87733435106002d54e11309700494e0e81","observation_id":"6c9239fd-3390-47f3-85c5-dfaf0914fc01","resolution":{"observed_at":"2026-08-14T14:14:49.208481Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.188011Z","title":"Modular forms in Pari/GP","venue":null,"work_id":"c7e31cf8-baf0-4740-934e-f71ad8c00d8f","year":2018},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.728088Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:b7411e9859ae3f7dff98c5fdb00dcc5fa3214a74a99f3f17bd51c8db158dd9e2","observation_id":"977e4849-b021-459d-ba88-abca4498e9cc","resolution":{"observed_at":"2026-08-14T14:14:49.192804Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.172782Z","title":"Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves","venue":null,"work_id":"a8ca32bc-faea-477c-88a2-16a388600e8b","year":2015},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.732907Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:ed4cfc540da67bd33f2129243cb3bc3de0fc86b37a37a5c4d6a2d18f3768bf89","observation_id":"d3291bc7-5641-4e75-9859-6b5582f779ff","resolution":{"observed_at":"2026-08-14T14:14:49.177855Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.158168Z","title":"A positive proportion of elliptic curves over Q have rank one","venue":null,"work_id":"8e30199b-cfba-495f-a2ed-314c555f4bbd","year":2014},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.737847Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:06d59e672b2d11e835d85ce4ab17a54083ed96e556e0d5b7140ed66b3caaaea2","observation_id":"66dc67f2-ea70-4679-af6d-e745ea156441","resolution":{"observed_at":"2026-08-14T14:14:49.162837Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.143281Z","title":"Toric modular forms and nonvanishing of L-functions","venue":null,"work_id":"59b5c299-efba-425c-b150-78074baf8883","year":2001},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.743539Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:30be2d82056a7b6994f263602bcd6d25323882db7f4520f5d7853ad726efb1db","observation_id":"66ec1ed7-9a64-4573-867a-9357e456a319","resolution":{"observed_at":"2026-08-14T14:14:49.148136Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.128239Z","title":"Toric modular forms of higher weight","venue":null,"work_id":"7c9e3392-bbe9-45d3-9129-82b980748ca7","year":2003},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.748979Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:1d23340dc830d63f2acbbb27c184aa321c87d2a4f95e1fd6106798e2e1a18975","observation_id":"708aa310-27f0-48d9-bd8b-18b51a5dcfeb","resolution":{"observed_at":"2026-08-14T14:14:49.133051Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.113103Z","title":"Toric varieties and modular forms","venue":null,"work_id":"236669ba-fbd5-4b6c-afbb-9d8a1ac62e31","year":2001},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.754614Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:4de33623400185c535a97a1354d9c399a158a236da043d510e41f6610c219de9","observation_id":"41a08edd-a7d5-45e2-9663-a8586248efc2","resolution":{"observed_at":"2026-08-14T14:14:49.118239Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.096781Z","title":"On two geometric theta lifts","venue":null,"work_id":"2c609bf9-1885-4e01-8ff2-d1abff0cd00d","year":2004},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.760472Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:42d22b420080e49b4a426e5abf6ceece03caa9f007fdb4309a070f2ba48e69d7","observation_id":"2b24a146-992a-4211-a993-785bb7dba7f8","resolution":{"observed_at":"2026-08-14T14:14:49.102273Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.081947Z","title":"Generalized moonshine, II: Borcherds products","venue":null,"work_id":"18117092-1513-41cc-b396-aa934341bed8","year":2012},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.764880Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:ae62304ce468035f765dbab105bc048babea640368787b13c1cd74454e480fad","observation_id":"cfb6b88d-be9d-4e85-bb6c-4e711f0b3029","resolution":{"observed_at":"2026-08-14T14:14:49.086700Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.066578Z","title":"Expansions at cusps and Petersson products in Pari/GP","venue":null,"work_id":"1773071f-1af0-4912-8753-18032a6160d6","year":2019},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.769179Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:a1618763cfa08037e263a7ce8d1cc43d3f1d635a62b45ce4c607bceda5f508c0","observation_id":"f82dbdb5-7cb1-4b6d-857d-0a5cefe91d79","resolution":{"observed_at":"2026-08-14T14:14:49.071907Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.052172Z","title":"La conjecture de Weil. II","venue":null,"work_id":"1793b603-aecb-4274-a2bd-526657226fe2","year":1980},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.773685Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:d716f864f44fdad69116817fa190ba72aabfbdc6484cbdbb503094d3849e1d03","observation_id":"88db78a8-c8a9-42ae-899e-8e68e7637427","resolution":{"observed_at":"2026-08-14T14:14:49.056787Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.037071Z","title":"Products of Eisenstein series and Fourier expansions of modular forms at cusps","venue":null,"work_id":"79b56d16-2ccd-4f96-8501-fff53a6b0ccd","year":2018},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.778180Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:50286c698acc936c8f60117b6d91154c41ff6731ee202c78dcbd6c6f34fa38e8","observation_id":"b5fc7675-aa5f-4de3-bfb9-f0fd0308b373","resolution":{"observed_at":"2026-08-14T14:14:49.042171Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.021422Z","title":"Nonvanishing theorems for automorphic L-functions on GL(2)","venue":null,"work_id":"32c5559c-c960-46ca-a922-f6a38a0cd34b","year":1995},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.783031Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:1933101b5b0875068d4ad68b29447797c23e70b92bc8923f39233c3a38fc484b","observation_id":"a93cc50f-a94e-4b85-a8ff-f9b28f63bc66","resolution":{"observed_at":"2026-08-14T14:14:49.026938Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:49.006064Z","title":"Conjectures on elliptic curves over quadratic ﬁelds","venue":null,"work_id":"69a2bda2-2b95-4c93-a5c3-ec2d1dcce22e","year":1979},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.787834Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:444f5c7c884ce7ba4959b67109a8b5229c78d2fa87cb7ed58839c0d953c58d7d","observation_id":"ec829c3f-5515-4c05-bdb2-34e1085c8471","resolution":{"observed_at":"2026-08-14T14:14:49.010848Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.990913Z","title":"Heegner points and derivatives of L-series","venue":null,"work_id":"967ea852-21e3-471a-a335-62831810e591","year":1986},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.792350Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:d2b800359238510e3b96fda50c5d4b74e4e495bd700d3f9e22d0b596b83268ba","observation_id":"9b886d49-72ab-437e-8de9-16f750ea465d","resolution":{"observed_at":"2026-08-14T14:14:48.995895Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.976410Z","title":"Iwaniec and E","venue":null,"work_id":"bc3af6b3-ed32-4a36-b81d-7541bd131bcd","year":2004},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.797335Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:f5cd20c491785f3d280dfe3c69409b406f1e3cb4b4d2216dd9c830f208e438ac","observation_id":"f280962c-bb75-4e2f-8526-6ac20e676b55","resolution":{"observed_at":"2026-08-14T14:14:48.980918Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.960860Z","title":"Products of two Eisenstein series and spaces of cusp forms of prime level","venue":null,"work_id":"3820f51b-b77e-4385-9c97-e2e592567d7f","year":2008},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.801794Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:cc05ae8fb4f2e87da75cbf92840f350c78f503082b83c2223aab3e27fd41cdc5","observation_id":"018eb2e0-52e7-47c3-82dd-37b5b45e03fc","resolution":{"observed_at":"2026-08-14T14:14:48.965898Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.944228Z","title":"Modular forms with rational periods","venue":null,"work_id":"faf0a1fe-b8ad-4c0b-affd-9c9ea47f3211","year":1983},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.806222Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:dec9d045d29e41a89e41f433c939f51d9930e233f2106768a5220774f525e634","observation_id":"a1c10b39-3e01-4c96-ac58-4a7efa3ceeba","resolution":{"observed_at":"2026-08-14T14:14:48.949744Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.927260Z","title":null,"venue":null,"work_id":"ee8f43b0-02a7-4bbf-93e0-1bb6c925a8c0","year":1989},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.810646Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:b02fe3a470c6f2d45c27af7c6ac4aedc1df61468a83c189e0f1b8c4b1bf5b2ea","observation_id":"39fa08bf-4f7c-4764-a4d1-1eca6c40daca","resolution":{"observed_at":"2026-08-14T14:14:48.932300Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.909784Z","title":"Non-vanishing of quadratic twists of modular L-functions","venue":null,"work_id":"0fd9c725-da39-4f54-ac50-8051dd0872ad","year":1998},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.815275Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:8045b9fdbc4fdd8cf51d33dbb96359a688653a6de350df8fa5b8f393bb42d067","observation_id":"8008cbc4-8276-4d7b-8e48-4c41e131296f","resolution":{"observed_at":"2026-08-14T14:14:48.915876Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.894534Z","title":"The scalar product of modular forms","venue":null,"work_id":"d42fe3a1-6669-479e-b751-624db1385bf8","year":1952},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.820023Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:ac536788560b2fe9a03c17f1cb889cf23c5a606fefa80d4b460d322e3cb08a04","observation_id":"a47b617f-5852-4380-b7f0-57cf8e34479b","resolution":{"observed_at":"2026-08-14T14:14:48.899439Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.878703Z","title":"Sur les coefﬁcients de Fourier des formes modulaires de poids demi-entier","venue":null,"work_id":"ab36f320-09da-433a-ba26-97ea28e9d956","year":1981},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.824789Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:97e6cb48f9daa5c50cb2ba0fa0ccdeac3fb1e261441d4f03f44123b29517e747","observation_id":"613bf0ea-e972-4864-a3c9-f5cc18cd508f","resolution":{"observed_at":"2026-08-14T14:14:48.883885Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T14:14:48.860297Z","title":"Products of vector valued Eisenstein series","venue":null,"work_id":"fc1ea182-b3d6-47a6-8f37-6ce58c705da3","year":2017},"citing_paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-14T14:14:48.829265Z"},"links":{"citing_paper":"/paper/1908.03616"},"observation_digest":"sha256:f301a0800eff3e4503150181746a8750d52ff26c7279c3b040d90f920155ec6a","observation_id":"1521ec6c-a455-4189-8e1b-f9ee11d2b52f","resolution":{"observed_at":"2026-08-14T14:14:48.867297Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"1908.03616","last_updated":"2019-08-09T20:12:07Z","latest_version":1,"primary_category":"math.NT","snapshot_observed_at":"2026-08-16T10:45:56.659153Z","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series"},"reference_resolution":{"displayed":23,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":1,"verified_exact":0,"verified_fuzzy":22},"total_outbound_references":23},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"thesis":"As of 16 August 2026, this Paper Citation Record lists 23 of 23 outbound references and 0 inbound Pith citation observations for arXiv:1908.03616."}