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Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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Supppose f and g are two normalized classical eigenforms of tame
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level N weight characters \kqppa and \tau on \Z_p^*. Suppose
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f(q)\con g(q) modulo p^m.
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Then, when $p$ is odd, I can show
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\kappa\con \tau modulo p^m.
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When f and g have level 1 and rational coefficients Serre showed the
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same is true when p=2, Thm. 1 of sect. 1.3 of Formes modulaires et
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fonctions zeta p-adic, but I don't believe it is true, in general, for
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p=2. I guess, I would first look at forms on X_1(4). (I am still
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trying to explain your computations of last Fall.)
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Robert
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