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gp:[N,k,chi] = [360,8,Mod(1,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 8, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,0,0,0,250,0,844]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = 8\sqrt{46}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(360))\):
\( T_{7}^{2} - 844T_{7} - 672732 \)
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|
\( T_{11}^{2} + 4640T_{11} + 1131264 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 125)^{2} \)
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|
| $7$ |
\( T^{2} - 844T - 672732 \)
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|
| $11$ |
\( T^{2} + 4640 T + 1131264 \)
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|
| $13$ |
\( T^{2} - 6804 T - 39722652 \)
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|
| $17$ |
\( T^{2} + 38516 T + 133419300 \)
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|
| $19$ |
\( T^{2} - 39720 T + 29646224 \)
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|
| $23$ |
\( T^{2} + 46244 T - 775126332 \)
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|
| $29$ |
\( T^{2} + \cdots - 4295905404 \)
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|
| $31$ |
\( T^{2} + \cdots + 11743009680 \)
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|
| $37$ |
\( T^{2} + \cdots - 236492620764 \)
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| $41$ |
\( T^{2} + \cdots - 1010797212 \)
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| $43$ |
\( T^{2} + \cdots + 196424013732 \)
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| $47$ |
\( T^{2} + \cdots + 150662471684 \)
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| $53$ |
\( T^{2} + \cdots - 2106147954876 \)
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|
| $59$ |
\( T^{2} + \cdots - 5676215308272 \)
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| $61$ |
\( T^{2} + \cdots - 2532614905500 \)
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| $67$ |
\( T^{2} + \cdots + 9716614914308 \)
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| $71$ |
\( T^{2} + \cdots + 285545583120 \)
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| $73$ |
\( T^{2} + \cdots + 2891492673348 \)
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| $79$ |
\( T^{2} + \cdots - 12100786389440 \)
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| $83$ |
\( T^{2} + \cdots + 28207563100452 \)
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| $89$ |
\( T^{2} + \cdots - 12795653008796 \)
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| $97$ |
\( T^{2} + \cdots - 44129117750396 \)
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