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gp:[N,k,chi] = [240,8,Mod(1,240)]
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("240.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,0,-54,0,-250,0,896]
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 \(\beta = 12\sqrt{10761}\).
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 kernel of the linear operator
\( T_{7}^{2} - 896T_{7} - 1348880 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T + 27)^{2} \)
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|
| $5$ |
\( (T + 125)^{2} \)
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|
| $7$ |
\( T^{2} - 896 T - 1348880 \)
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|
| $11$ |
\( T^{2} - 5152 T - 18157568 \)
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|
| $13$ |
\( T^{2} - 8396 T - 58306412 \)
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|
| $17$ |
\( T^{2} - 7780 T - 172367564 \)
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|
| $19$ |
\( T^{2} + 54656 T + 732873328 \)
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|
| $23$ |
\( T^{2} + \cdots + 3292226560 \)
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|
| $29$ |
\( T^{2} + \cdots - 6506543996 \)
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|
| $31$ |
\( T^{2} + \cdots - 30482359040 \)
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|
| $37$ |
\( T^{2} + \cdots - 25239909420 \)
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|
| $41$ |
\( T^{2} + \cdots + 21672667908 \)
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|
| $43$ |
\( T^{2} + \cdots + 1796873104 \)
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|
| $47$ |
\( T^{2} + \cdots + 62465213440 \)
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|
| $53$ |
\( T^{2} + \cdots - 1218461818140 \)
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|
| $59$ |
\( T^{2} + \cdots - 1928491876352 \)
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|
| $61$ |
\( T^{2} + \cdots + 3435987098788 \)
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|
| $67$ |
\( T^{2} + \cdots - 4575294566768 \)
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|
| $71$ |
\( T^{2} + \cdots - 458660708352 \)
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|
| $73$ |
\( T^{2} + \cdots + 8033732644644 \)
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|
| $79$ |
\( T^{2} + \cdots - 2328110775936 \)
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|
| $83$ |
\( T^{2} + \cdots - 25191624019952 \)
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|
| $89$ |
\( T^{2} + \cdots - 42436810426140 \)
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|
| $97$ |
\( T^{2} + \cdots - 26674357148156 \)
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