The only rational weights k for which the KZ equation has modular form solutions on Γ(N) are those with k ≡ 1/2, 7/2, 1, 2, 3 mod 6 or k = (6n+1)/5.
Classical and quantum conformal field theory.Communications in Mathematical Physics, 123(2):177–254
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On modular forms of rational weight satisfying the canonical second-order linear modular differential equation
The only rational weights k for which the KZ equation has modular form solutions on Γ(N) are those with k ≡ 1/2, 7/2, 1, 2, 3 mod 6 or k = (6n+1)/5.