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Fourier coefficients of trace operator on modular forms
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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 $SL_2(mathbbZ)$ by
$ (Tr_1^D f)(z) = sumlimits_beginpmatrix a &b\c & d endpmatrixin Gamma_0(D)/SL_2mathbbZ (cz+d)^-kfleft(fracaz+bcz+dright); $ where $beginpmatrix a &b\c & d endpmatrix$ 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 $(fracD.) $ 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?
modular-forms
asked Jul 30 at 10:35
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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 $SL_2(mathbbZ)$ by
$ (Tr_1^D f)(z) = sumlimits_beginpmatrix a &b\c & d endpmatrixin Gamma_0(D)/SL_2mathbbZ (cz+d)^-kfleft(fracaz+bcz+dright); $ where $beginpmatrix a &b\c & d endpmatrix$ 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 $(fracD.) $ 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?
modular-forms
asked Jul 30 at 10:35
1.414212
133
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 $SL_2(mathbbZ)$ by
$ (Tr_1^D f)(z) = sumlimits_beginpmatrix a &b\c & d endpmatrixin Gamma_0(D)/SL_2mathbbZ (cz+d)^-kfleft(fracaz+bcz+dright); $ where $beginpmatrix a &b\c & d endpmatrix$ 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 $(fracD.) $ 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?
modular-forms
asked Jul 30 at 10:35
1.414212
133
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 $SL_2(mathbbZ)$ by
$ (Tr_1^D f)(z) = sumlimits_beginpmatrix a &b\c & d endpmatrixin Gamma_0(D)/SL_2mathbbZ (cz+d)^-kfleft(fracaz+bcz+dright); $ where $beginpmatrix a &b\c & d endpmatrix$ 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 $(fracD.) $ 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?
modular-forms
asked Jul 30 at 10:35
1.414212
133
asked Jul 30 at 10:35
1.414212
133
asked Jul 30 at 10:35
1.414212
133
asked Jul 30 at 10:35
1.414212
133
133
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add a comment |Â
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