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gp:[N,k,chi] = [25,8,Mod(1,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,-15]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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sage:f.q_expansion() # note that sage often uses an isomorphic number field
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gp:mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{649})\).
We also show the integral \(q\)-expansion of the trace form.
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.
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gp:mfembed(f)
| \( p \) |
Sign
|
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 15T_{2} - 106 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(25))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 15T - 106 \)
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|
| $3$ |
\( T^{2} + 40T - 249 \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 600 T - 1054836 \)
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|
| $11$ |
\( T^{2} - 4344 T - 5423041 \)
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|
| $13$ |
\( T^{2} + 17680 T + 51136816 \)
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|
| $17$ |
\( T^{2} + 6870 T - 68614471 \)
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|
| $19$ |
\( T^{2} - 18200 T - 117098225 \)
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|
| $23$ |
\( T^{2} + \cdots - 7241627844 \)
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|
| $29$ |
\( T^{2} + \cdots - 27320953600 \)
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|
| $31$ |
\( T^{2} + \cdots + 19481626044 \)
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|
| $37$ |
\( T^{2} + \cdots + 72425629684 \)
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|
| $41$ |
\( T^{2} + \cdots + 2293303049 \)
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|
| $43$ |
\( T^{2} + \cdots + 168009863536 \)
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|
| $47$ |
\( T^{2} + \cdots + 768835854224 \)
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|
| $53$ |
\( T^{2} + \cdots - 310528480924 \)
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|
| $59$ |
\( T^{2} + \cdots - 174052062400 \)
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|
| $61$ |
\( T^{2} + \cdots - 8129212445516 \)
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|
| $67$ |
\( T^{2} + \cdots - 858879415521 \)
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|
| $71$ |
\( T^{2} + \cdots + 5183381635664 \)
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|
| $73$ |
\( T^{2} + \cdots - 708123554519 \)
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|
| $79$ |
\( T^{2} + \cdots + 11646468081900 \)
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|
| $83$ |
\( T^{2} + \cdots + 35517015006471 \)
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|
| $89$ |
\( T^{2} + \cdots - 22736713427775 \)
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|
| $97$ |
\( T^{2} + \cdots + 70829616805924 \)
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