Function: mfshimura
Section: modular_forms
C-Name: mfshimura
Prototype: GGD1,L,
Help: mfshimura(mf, F, {D = 1}): F being a modular form of
half-integral weight k >= 3/2 and D a (not necessarily fundamental)
discriminant of suitable sign, computes the Shimura lift G of weight 2k-1
corresponding to D. This function returns [mf2,G,v], where mf2 is a modular
form space containing G, and v the vector of coefficients of G on mf. By
extension, D can also be a positive squarefree integer.
Doc: $F$ being a modular form of half-integral weight $k\geq 3/2$ and $D$ a
(not necessarily fundamental) discriminant of suitable sign, returns the
Shimura lift $G$ of weight $2k-1$ corresponding to $D$. The sign of $D$
should be equal to $(-1)^{k-1/2}\epsilon$, where $\epsilon=1$ if $\chi$ is
defined modulo $N/4$, $\epsilon=-1$ if $(-4/.)\chi$ is defined modulo
$N/4$ (and error otherwise). This function returns $[\var{mf2},G,v]$
where \var{mf2} is a modular form space containing $G$ and $v$ expresses $G$
in terms of \kbd{mfbasis}$(\var{mf2})$; so that $G$ is
\kbd{mflinear}$(\var{mf2},v)$. By extension, we allow $D$ to be a positive
squarefree integer, but in that case if $(-1)^{k-1/2}\epsilon\cdot D$ is not a
discriminant, the Kohnen $+$-space will in general not be sent to
level $N/4$ but only $N/2$.
\bprog
? F = mfpow(mfTheta(), 7); mf = mfinit(F);
? [mf2, G, v] = mfshimura(mf, F, -3); mfcoefs(G,5)
%2 = [-5/9, 280, 9240, 68320, 295960, 875280]
? mfparams(G)
%3 = [1, 6, 1, y]
? mfparams(mf2)
%4 = [2, 6, 1, 4] \\ it may happen that G has lower level than expected
? v
%5 = [280, 0]~
? mfcoefs(mf2, 5)
%6 =
[-1/504 -1/504]
[ 1 0]
[ 33 1]
[ 244 0]
[ 1057 33]
[ 3126 0]
? mf = mfinit([60,5/2],1); F = mflinear(mf,mfkohnenbasis(mf)[,1]);
? mfparams(mfshimura(mf,F,1)[2])
%8 = [15, 4, 1, y]
? mfparams(mfshimura(mf,F,6)[2])
%9 = [30, 4, 1, y]
@eprog