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I am currently reading the paper Values of L-series of Modular Forms at the Center of the Critical Strip (https://eudml.org/doc/142819). In page 185( just before prop. 1), $ Tr_N^M$ denotes the trace operator (adjoint to the inclusion map) from modular forms on $\Gamma_0(M)$ to forms on $\Gamma_0(N)$. In page 194, the last paragraph defines the trace operator from $\Gamma_0(D)$ to $SL_2(\mathbb{Z})$ by

$ (Tr_1^D f)(z) = \sum\limits_{\begin{pmatrix} a &b\\c & d \end{pmatrix}\in \Gamma_0(D)/SL_2\mathbb{Z}} (cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right); $ where $\begin{pmatrix} a &b\\c & d \end{pmatrix}$ runs over the representatives. The paper then gives the Fourier coefficients of a modular form which is, trace applied on the product of a certain Eisenstein series with character $(\frac{D}{.}) $ and the theta series.

My questions:

1) How does the formula looks for general $M$ & $N$? (I couldn't find any proper reference in any book or online) - Is it just varying the matrix over all the representatives in this case also?

2) I am trying to understand the Fourier series computations in the appendix, which is tricky. So, I am trying this way - first compute the Fourier coefficients of a) $Tr_N^M$(Eisenstein Series) b) $Tr_N^M$(Theta series); and then find the product(which I can do by Cauchy product). Is this approach correct? If so, how to find the Fourier coefficients of the trace of a modular form whose coefficients are known?

3) In my understanding, coefficient calculation depends on the choice of representatives. Is there any particular way of selecting the choice?

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    $\begingroup$ Without putting much thought into it, I would say: 1) yes, 3) it should not depend upon the choice) 2) use the definition. $\endgroup$
    – Kimball
    Dec 21, 2018 at 15:33
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    $\begingroup$ Regarding 2) there is no general formula for the Fourier coeffs of the trace of a general modular form. This is because the elements of $\Gamma_0 (N) $ used in the definition of the trace will move the infinity cusp to various other cusps. So the right approach instead is to compute the slash operators on each factor of the product. This works here because the action of slash operators on Eisenstein series and on the theta series considered here are known. $\endgroup$ Dec 21, 2018 at 19:13
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    $\begingroup$ Also, in 2) it seems you want to take the product of the traces of the modular forms, but this is not the same as the trace of a product. $\endgroup$ Dec 21, 2018 at 20:56
  • $\begingroup$ @FrançoisBrunault 2) So product of traces and trace of products, are both different? And, I will try your method and write the answer. $\endgroup$
    – 1.414212
    Dec 22, 2018 at 16:52
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    $\begingroup$ @1.414212 Yes, the trace maps do not respect the product structure on modular forms. Similarly for the Hecke operators. Looking at the definitions, it is pretty clear that there is no reason for this to hold. $\endgroup$ Dec 22, 2018 at 18:23

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