[N,k,chi] = [2624,1,Mod(97,2624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
chi = DirichletCharacter(H, H._module([0, 20, 37]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2624.97");
S:= CuspForms(chi, 1);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).
\(n\) |
\(129\) |
\(575\) |
\(1477\) |
\(\chi(n)\) |
\(-\zeta_{40}^{13}\) |
\(1\) |
\(-1\) |
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 2 T_{3}^{14} - 4 T_{3}^{13} + 2 T_{3}^{12} + 16 T_{3}^{11} + 16 T_{3}^{10} - 52 T_{3}^{9} + 75 T_{3}^{8} - 72 T_{3}^{7} + 28 T_{3}^{6} - 52 T_{3}^{5} + 82 T_{3}^{4} + 136 T_{3}^{3} + 58 T_{3}^{2} + 8 T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2624, [\chi])\).
$p$ |
$F_p(T)$ |
$2$ |
\( T^{16} \)
|
$3$ |
\( T^{16} - 2 T^{14} - 4 T^{13} + 2 T^{12} + \cdots + 1 \)
|
$5$ |
\( T^{16} \)
|
$7$ |
\( T^{16} \)
|
$11$ |
\( T^{16} - 4 T^{15} + 10 T^{14} - 20 T^{13} + \cdots + 1 \)
|
$13$ |
\( T^{16} \)
|
$17$ |
\( T^{16} + 4 T^{15} + 10 T^{14} + 20 T^{13} + \cdots + 16 \)
|
$19$ |
\( T^{16} + 16 T^{15} + 120 T^{14} + 560 T^{13} + \cdots + 1 \)
|
$23$ |
\( T^{16} \)
|
$29$ |
\( T^{16} \)
|
$31$ |
\( T^{16} \)
|
$37$ |
\( T^{16} \)
|
$41$ |
\( T^{16} - T^{12} + T^{8} - T^{4} + 1 \)
|
$43$ |
\( T^{16} + 5 T^{12} + 150 T^{8} + \cdots + 625 \)
|
$47$ |
\( T^{16} \)
|
$53$ |
\( T^{16} \)
|
$59$ |
\( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \)
|
$61$ |
\( T^{16} \)
|
$67$ |
\( T^{16} + 8 T^{14} - 4 T^{13} + 27 T^{12} + \cdots + 1 \)
|
$71$ |
\( T^{16} \)
|
$73$ |
\( (T^{8} + 7 T^{4} + 1)^{2} \)
|
$79$ |
\( T^{16} \)
|
$83$ |
\( (T^{4} + 5 T^{2} + 5)^{4} \)
|
$89$ |
\( T^{16} - 4 T^{15} + 10 T^{14} - 20 T^{13} + \cdots + 1 \)
|
$97$ |
\( T^{16} - 2 T^{14} + 16 T^{13} + 2 T^{12} + \cdots + 1 \)
|
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