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Computational Aspects of Modular Forms and Galois Representations by Robin de Jong, Jean-Marc Couveignes, Franz Merkl, Johan Bosman, Bas Edixhoven

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So all the monomials (Image(m))a j–b arising in the equation E(Image(m), j−1) = 0 have exp-type

l2 (m + 1), Θl2(l + m + 1)2, 4),

and the coefficients in their q-expansions up to order

2 deg(Image(m)) × deg(j–1 ≤ Θ(m + 1)l4

are rational integers bounded in absolute value by

exp(Θl24(m + 1)8).

Since the coefficients in E(x, y) are solutions of the homogeneous system given by these truncated q-expansions, they are bounded in absolute value by

exp(Θl28(m + 1)9).

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