Sums of absolute Hecke eigenvalues for GL(2) representations exhibit logarithmic savings over trivial bounds if and only if the representation is cuspidal, with a connection drawn to base change.
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Sums of Hecke eigenvalues along polynomial sequences and base change for $\text{GL}(2)$
Sums of absolute Hecke eigenvalues for GL(2) representations exhibit logarithmic savings over trivial bounds if and only if the representation is cuspidal, with a connection drawn to base change.