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gp:[N,k,chi] = [96,10,Mod(1,96)]
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("96.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(96, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
Newform invariants
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sage:traces = [2,0,-162,0,596]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\beta = 48\sqrt{46}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( +1 \) |
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 \(S_{10}^{\mathrm{new}}(\Gamma_0(96))\):
\( T_{5}^{2} - 596T_{5} - 17180 \)
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|
\( T_{7}^{2} - 5304T_{7} - 23596272 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T + 81)^{2} \)
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|
| $5$ |
\( T^{2} - 596T - 17180 \)
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|
| $7$ |
\( T^{2} - 5304 T - 23596272 \)
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|
| $11$ |
\( T^{2} + \cdots - 2776206960 \)
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|
| $13$ |
\( T^{2} + \cdots - 25843390524 \)
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|
| $17$ |
\( T^{2} + \cdots - 62209429916 \)
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|
| $19$ |
\( T^{2} + \cdots - 116097099120 \)
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|
| $23$ |
\( T^{2} + \cdots - 1525603237632 \)
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|
| $29$ |
\( T^{2} + \cdots - 5606206497500 \)
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|
| $31$ |
\( T^{2} + \cdots - 21456001704816 \)
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|
| $37$ |
\( T^{2} + \cdots - 12311178604412 \)
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|
| $41$ |
\( T^{2} + \cdots - 150213628799804 \)
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| $43$ |
\( T^{2} + \cdots + 455488413078672 \)
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| $47$ |
\( T^{2} + \cdots + 589985752076352 \)
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| $53$ |
\( T^{2} + \cdots + 25\!\cdots\!80 \)
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|
| $59$ |
\( T^{2} + \cdots + 60\!\cdots\!92 \)
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| $61$ |
\( T^{2} + \cdots + 14\!\cdots\!80 \)
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| $67$ |
\( T^{2} + \cdots + 30\!\cdots\!20 \)
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| $71$ |
\( T^{2} + \cdots + 36\!\cdots\!08 \)
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| $73$ |
\( T^{2} + \cdots - 26\!\cdots\!20 \)
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| $79$ |
\( T^{2} + \cdots + 33\!\cdots\!12 \)
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| $83$ |
\( T^{2} + \cdots + 20\!\cdots\!76 \)
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| $89$ |
\( T^{2} + \cdots + 13\!\cdots\!76 \)
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|
| $97$ |
\( T^{2} + \cdots - 39\!\cdots\!24 \)
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