mfeisenstein

Function: mfeisenstein
Section: modular_forms
C-Name: mfeisenstein
Prototype: LDGDG
Help: mfeisenstein(k,{CHI1},{CHI2}): create the Eisenstein
 E_k(CHI1,CHI2), where an omitted character is considered as trivial.
Doc: create the Eisenstein series $E_k(\chi_1,\chi_2)$, where $k \geq 1$,
 $\chi_i$ are Dirichlet characters and an omitted character is considered as
 trivial. This form belongs to ${\cal E}_k(\Gamma_0(N), \chi)$ with $\chi =
 \chi_1\chi_2$ and $N$ is the product of the conductors of $\chi_1$ and
 $\chi_2$.
 \bprog
 ? CHI = Mod(3,4);
 ? E = mfeisenstein(3, CHI);
 ? mfcoefs(E, 6)
 %2 = [-1/4, 1, 1, -8, 1, 26, -8]
 ? CHI2 = Mod(4,5);
 ? mfcoefs(mfeisenstein(3,CHI,CHI2), 6)
 %3 = [0, 1, -1, -10, 1, 25, 10]
 ? mfcoefs(mfeisenstein(4,CHI,CHI), 6)
 %4 = [0, 1, 0, -28, 0, 126, 0]
 ? mfcoefs(mfeisenstein(4), 6)
 %5 = [1/240, 1, 9, 28, 73, 126, 252]
 @eprog\noindent Note that \kbd{mfeisenstein}$(k)$ is 0 for $k$ odd and
 $-B_{k}/(2k) \cdot E_k$ for $k$ even, where
 $$E_k(q) = 1 - (2k/B_k)\sum_{n\geq 1} \sigma_{k-1}(n) q^n$$
 is the standard Eisenstein series. In other words it is normalized so that its
 linear coefficient is $1$.

 \misctitle{Important note} This function is currently implemented only when
 $\Q(\chi)$ is the field of definition of $E_k(\chi_1,\chi_2)$. If it is a
 strict subfield, an error is raised:
 \bprog
 ? mfeisenstein(6, Mod(7,9), Mod(4,9));
  ***   at top-level: mfeisenstein(6,Mod(7,9),Mod(4,9))
  ***                 ^---------------------------------
  *** mfeisenstein: sorry, mfeisenstein for these characters is not
  *** yet implemented.
 @eprog\noindent The reason for this is that each modular form is attached
 to a modular form space $M_k(\Gamma_0(N),\chi)$. This is a $\C$-vector
 space but it allows a basis of forms defined over $\Q(\chi)$ and is only
 implemented as a $\Q(\chi)$-vector space: there is
 in general no mechanism to take linear combinations of forms in the space
 with coefficients belonging to a larger field. (Due to their importance,
 eigenforms are the single exception to this restriction; for an eigenform
 $F$, $\Q(F)$ is built on top of $\Q(\chi)$.) When the property $\Q(\chi) =
 \Q(E_k(\chi_1,\chi_2)$ does not hold, we cannot express $E$ as a
 $\Q(\chi)$-linear combination of the basis forms and many operations will
 fail. For this reason, the construction is currently disabled.