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gp:[N,k,chi] = [2312,2,Mod(1,2312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [3,0,3,0,-6,0,3,0,0,0,6,0,0,0,-9]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == 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
|
| \(2\) |
\( -1 \) |
| \(17\) |
\( +1 \) |
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\):
\( T_{3}^{3} - 3T_{3}^{2} + 1 \)
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\( T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 1 \)
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\( T_{7}^{3} - 3T_{7}^{2} - 18T_{7} + 37 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} \)
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|
| $3$ |
\( T^{3} - 3T^{2} + 1 \)
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|
| $5$ |
\( T^{3} + 6 T^{2} + \cdots + 1 \)
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|
| $7$ |
\( T^{3} - 3 T^{2} + \cdots + 37 \)
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|
| $11$ |
\( T^{3} - 6 T^{2} + \cdots - 3 \)
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|
| $13$ |
\( T^{3} - 21T - 17 \)
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|
| $17$ |
\( T^{3} \)
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|
| $19$ |
\( T^{3} + 6 T^{2} + \cdots - 213 \)
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|
| $23$ |
\( T^{3} + 3 T^{2} + \cdots - 57 \)
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|
| $29$ |
\( T^{3} - 3 T^{2} + \cdots - 1 \)
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|
| $31$ |
\( T^{3} - 12 T^{2} + \cdots - 19 \)
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| $37$ |
\( T^{3} - 18 T^{2} + \cdots - 171 \)
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| $41$ |
\( T^{3} - 9T^{2} + 81 \)
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| $43$ |
\( T^{3} - 15 T^{2} + \cdots - 109 \)
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| $47$ |
\( T^{3} + 6 T^{2} + \cdots - 969 \)
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| $53$ |
\( T^{3} + 9 T^{2} + \cdots - 323 \)
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|
| $59$ |
\( T^{3} - 3 T^{2} + \cdots - 1 \)
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| $61$ |
\( T^{3} - 3 T^{2} + \cdots - 37 \)
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| $67$ |
\( T^{3} + 6 T^{2} + \cdots - 1369 \)
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| $71$ |
\( T^{3} - 9 T^{2} + \cdots + 323 \)
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| $73$ |
\( T^{3} - 12 T^{2} + \cdots + 73 \)
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| $79$ |
\( T^{3} + 6 T^{2} + \cdots + 8 \)
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| $83$ |
\( T^{3} - 21 T^{2} + \cdots + 2557 \)
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| $89$ |
\( T^{3} - 18 T^{2} + \cdots + 631 \)
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| $97$ |
\( T^{3} + 9 T^{2} + \cdots - 37 \)
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