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A criterion is given for a function to be a modular form for      Γ  0 ( N ) {\Gamma _0}(N)      . It is similar to the criterion given by Weil in his 1967 Math. Ann. paper  Über die Bestimmung Dirichletscher Reihen durch Funktional - gleichungen  in that it involves checking that certain twists of the associated Dirichlet series satisfy functional equations. It differs in the number and type of such equations which need to be satisfied.

Modular forms for 𝐺₀(𝑁) and Dirichlet series