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gp:[N,k,chi] = [1360,4,Mod(1,1360)]
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("1360.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [3,0,4,0,-15,0,-8,0,39,0,118]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
| \(17\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1360))\):
\( T_{3}^{3} - 4T_{3}^{2} - 52T_{3} + 64 \)
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\( T_{7}^{3} + 8T_{7}^{2} - 1028T_{7} - 13648 \)
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\( T_{11}^{3} - 118T_{11}^{2} + 3764T_{11} - 31128 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} - 4 T^{2} + \cdots + 64 \)
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|
| $5$ |
\( (T + 5)^{3} \)
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|
| $7$ |
\( T^{3} + 8 T^{2} + \cdots - 13648 \)
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|
| $11$ |
\( T^{3} - 118 T^{2} + \cdots - 31128 \)
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|
| $13$ |
\( T^{3} + 10 T^{2} + \cdots - 4808 \)
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| $17$ |
\( (T - 17)^{3} \)
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| $19$ |
\( T^{3} - 160 T^{2} + \cdots + 1236992 \)
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| $23$ |
\( T^{3} - 92 T^{2} + \cdots + 37824 \)
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| $29$ |
\( T^{3} - 374 T^{2} + \cdots + 1059384 \)
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| $31$ |
\( T^{3} + 70 T^{2} + \cdots + 869848 \)
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| $37$ |
\( T^{3} + 246 T^{2} + \cdots - 107032 \)
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| $41$ |
\( T^{3} + 354 T^{2} + \cdots + 268056 \)
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| $43$ |
\( T^{3} - 286 T^{2} + \cdots + 3687392 \)
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| $47$ |
\( T^{3} - 390 T^{2} + \cdots + 1611552 \)
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| $53$ |
\( T^{3} + 1282 T^{2} + \cdots - 35263752 \)
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| $59$ |
\( T^{3} - 304 T^{2} + \cdots + 121786368 \)
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| $61$ |
\( T^{3} - 246 T^{2} + \cdots + 59018936 \)
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| $67$ |
\( T^{3} - 718 T^{2} + \cdots + 59593088 \)
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| $71$ |
\( T^{3} - 602 T^{2} + \cdots + 23858136 \)
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| $73$ |
\( T^{3} - 282 T^{2} + \cdots + 187092392 \)
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| $79$ |
\( T^{3} + 206 T^{2} + \cdots - 3736296 \)
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| $83$ |
\( T^{3} - 2414 T^{2} + \cdots - 270648096 \)
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| $89$ |
\( T^{3} + 870 T^{2} + \cdots - 7132536 \)
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| $97$ |
\( T^{3} - 2230 T^{2} + \cdots + 724979000 \)
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