mfinit

Function: mfinit
Section: modular_forms
C-Name: mfinit
Prototype: GD4,L,
Help: mfinit(NK,{space=4}): Create the space of modular forms corresponding
 to the data contained in NK and space. NK is a vector which can be
 either [N,k] (N level, k weight) corresponding to a subspace of M_k(G_0(N)),
 or [N,k,CHI] (CHI a character) corresponding to a subspace of M_k(G_0(N),chi).
 The subspace is described by a small integer 'space': 0 for the newspace,
 1 for the cuspidal space, 2 for the oldspace, 3 for the space of Eisenstein
 series and 4 (default) for the full space M_k
Doc: Create the space of modular forms corresponding to the data contained in
 \kbd{NK} and \kbd{space}. \kbd{NK} is a vector which can be
 either $[N,k]$ ($N$ level, $k$ weight) corresponding to a subspace of
 $M_k(\Gamma_0(N))$, or $[N,k,\var{CHI}]$ (\var{CHI} a character)
 corresponding to a subspace of $M_k(\Gamma_0(N),\chi)$. Alternatively,
 it can be a modular form $F$ or modular form space, in which case we use
 \kbd{mfparams} to define the space parameters.

 The subspace is described by the small integer \kbd{space}: $0$ for the
 newspace $S_k^{\text{new}}(\Gamma_0(N),\chi)$, $1$ for the cuspidal
 space $S_k$, $2$ for the oldspace $S_k^{\text{old}}$, $3$ for the space of
 Eisenstein series $E_k$ and $4$ for the full space $M_k$.

 \misctitle{Wildcards} For given level and weight, it is advantageous to
 compute simultaneously spaces attached to different Galois orbits
 of characters, especially in weight $1$. The parameter \var{CHI} may be set
 to 0 (wildcard), in which case we return a vector of all \kbd{mfinit}(s) of
 non trivial spaces in $S_k(\Gamma_1(N))$, one for each Galois orbit
 (see \kbd{znchargalois}). One may also set \var{CHI} to a vector of
 characters and we return a vector of all mfinits of subspaces of
 $M_k(G_0(N),\chi)$ for $\chi$ in the list, in the same order. In weight $1$,
 only $S_1^{\text{new}}$, $S_1$ and $E_1$ support wildcards.

 The output is a technical structure $S$, or a vector of structures if
 \var{CHI} was a wildcard, which contains the following information:
 $[N,k,\chi]$ is given by \kbd{mfparams}$(S)$, the space
 dimension is \kbd{mfdim}$(S)$ and a $\C$-basis for the space is
 \kbd{mfbasis}$(S)$. The structure is entirely algebraic and does not depend
 on the current \kbd{realbitprecision}.
 \bprog
 ? S = mfinit([36,2], 0); \\ new space
 ? mfdim(S)
 %2 = 1
 ? mfparams
 %3 = [36, 2, 1, y]  \\ trivial character
 ? f = mfbasis(S)[1]; mfcoefs(f,10)
 %4 = [0, 1, 0, 0, 0, 0, 0, -4, 0, 0, 0]

 ? vS = mfinit([36,2,0],0); \\ with wildcard
 ? #vS
 %6 = 4   \\ 4 non trivial spaces (mod Galois action)
 ? apply(mfdim,vS)
 %7 = [1, 2, 1, 4]
 ? mfdim([36,2,0], 0)
 %8 = [[1, Mod(1, 36), 1, 0], [2, Mod(35, 36), 2, 0], [3, Mod(13, 36), 1, 0],
       [6, Mod(11, 36), 4, 0]]
 @eprog