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gp:[N,k,chi] = [234,6,Mod(1,234)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(234, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("234.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-8,0,32,56]
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 \(\beta = 2\sqrt{94}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( +1 \) |
| \(13\) |
\( -1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(234))\):
\( T_{5}^{2} - 56T_{5} + 408 \)
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\( T_{7}^{2} + 60T_{7} - 17524 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 56T + 408 \)
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|
| $7$ |
\( T^{2} + 60T - 17524 \)
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|
| $11$ |
\( T^{2} - 68T - 233844 \)
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|
| $13$ |
\( (T - 169)^{2} \)
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|
| $17$ |
\( T^{2} - 1496 T + 264720 \)
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|
| $19$ |
\( T^{2} + 540T - 288436 \)
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|
| $23$ |
\( T^{2} - 4536 T + 758160 \)
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| $29$ |
\( T^{2} - 2904 T - 7042032 \)
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| $31$ |
\( T^{2} + 6236 T + 9556108 \)
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| $37$ |
\( T^{2} + 15516 T + 35545028 \)
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| $41$ |
\( T^{2} - 8864 T - 83204280 \)
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| $43$ |
\( T^{2} + 1200 T - 392365984 \)
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| $47$ |
\( T^{2} - 14732 T + 40541100 \)
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| $53$ |
\( T^{2} - 11264 T + 28397088 \)
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|
| $59$ |
\( T^{2} + \cdots + 1082835084 \)
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|
| $61$ |
\( T^{2} - 11772 T - 698109820 \)
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| $67$ |
\( T^{2} + \cdots + 1002584908 \)
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| $71$ |
\( T^{2} + \cdots - 1518230388 \)
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| $73$ |
\( T^{2} - 35404 T + 625060 \)
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| $79$ |
\( T^{2} + \cdots - 4189595504 \)
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| $83$ |
\( T^{2} + \cdots + 2693685996 \)
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| $89$ |
\( T^{2} + \cdots - 7363927992 \)
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| $97$ |
\( T^{2} + \cdots - 16086584860 \)
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