Suppose you have a modular form of integer weight with integer Fourier coefficients that you suspect to be not congruent to zero modulo a fixed prime number. How can we prove that it is not ?
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2$\begingroup$ There is an upper bound, say N, on the number of zeros counted with multiplicity, which works in char p which depends on the weight (and level if you are not just talking about level one). So, if the first N+1 coefficients are divisible by a prime p, the form has too high a zero at the cusp so is identically zero modulo p. $\endgroup$– Felipe VolochCommented Mar 6, 2017 at 20:31
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1$\begingroup$ @FelipeVoloch I interpreted the question as asking how to show a specific $a_n$ (which I guess one can't compute?) is non-zero mod $p$. The OP should clarify the question. $\endgroup$– KimballCommented Mar 6, 2017 at 23:02
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I suspect that Felipe's interpretation is what you had in mind. I know that he's reluctant to post his excellent answers (he once helped me with a very useful comment which he wouldn't post as an answer) so I'll augment the comment above with an appropriate reference:
Springer Lecture Notes in Mathematics v.1240, "On the Congruence of Modular Forms"---Jacob Sturm, Theorem 1 on page 276.
answered Mar 28, 2017 at 0:27