mfdim

Function: mfdim
Section: modular_forms
C-Name: mfdim
Prototype: GD4,L,
Help: mfdim(NK,{space=4}): If NK=[N,k,CHI] as in
 mfinit, gives the dimension of the corresponding 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.
 NK can also be the output of mfinit, in which case space must be omitted.
Doc: If $NK=[N,k,\var{CHI}]$ as in \kbd{mfinit}, gives the dimension of the
 corresponding subspace of $M_k(\Gamma_0(N),\chi)$. $NK$ can also be the
 output of \kbd{mfinit}, in which case space must be omitted.

 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}
 As in \kbd{mfinit}, \var{CHI} may be the wildcard 0
 (all Galois orbits of characters); in this case, the output is a vector of
 $[\var{order}, \var{conrey}, \var{dim}, \var{dimdih}]$ corresponding
 to the nontrivial spaces, where

 \item \var{order} is the order of the character,

 \item \var{conrey} its Conrey label from which the character may be recovered
 via \kbd{znchar}$(\var{conrey})$,

 \item \var{dim} the dimension of the corresponding space,

 \item \var{dimdih} the dimension of the subspace of dihedral forms
 corresponding to Hecke characters if $k = 1$ (this is not implemented for
 the old space and set to $-1$ for the time being) and 0 otherwise.

 The spaces are sorted by increasing order of the character; the characters are
 taken up to Galois conjugation and the Conrey number is the minimal one among
 Galois conjugates. In weight $1$, this is only implemented when
 the space is 0 (newspace), 1 (cusp space), 2(old space) or 3(Eisenstein
 series).

 \misctitle{Wildcards for sets of characters} \var{CHI} may be a set
 of characters, and we return the set of $[\var{dim},\var{dimdih}]$.

 \misctitle{Wildcard for $M_k(\Gamma_1(N))$}
 Additionally, the wildcard $\var{CHI} = -1$ is available in which case we
 output the total dimension of the corresponding
 subspace of $M_k(\Gamma_1(N))$. In weight $1$, this is not implemented
 when the space is 4 (fullspace).

 \bprog
 ? mfdim([23,2], 0) \\ new space
 %1 = 2
 ? mfdim([96,6], 0)
 %2 = 10
 ? mfdim([10^9,4], 3)  \\ Eisenstein space
 %1 = 40000
 ? mfdim([10^9+7,4], 3)
 %2 = 2
 ? mfdim([68,1,-1],0)
 %3 = 3
 ? mfdim([68,1,0],0)
 %4 = [[2, Mod(67, 68), 1, 1], [4, Mod(47, 68), 1, 1]]
 ? mfdim([124,1,0],0)
 %5 = [[6, Mod(67, 124), 2, 0]]
 @eprog
 This last example shows that there exists a nondihedral form of weight 1
 in level 124.