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gp:[N,k,chi] = [9747,2,Mod(1,9747)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9747, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9747.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [3,3,0,3,-3,0,-9,6,0,0,-9,0,9,-12,0,3,-3,0,0,-3,0,-18,3,0,-6,
12,0,-18,6,0,0,0,0,9,12,0,27,0,0,-18,21,0,-18,-18,0,9,-6,0,12,-15,0,9,
6,0,0,-33,0,-9,3,0,-9,3,0,12,-3,0,9,33,0,9,3,0,-18,21,0,0,27,0,0,-12,0,
18,0,0,-9,-3,0,-9,33,0,-24]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == 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
|
| \(3\) |
\( +1 \) |
| \(19\) |
\( -1 \) |
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9747))\):
\( T_{2}^{3} - 3T_{2}^{2} + 3 \)
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\( T_{5}^{3} + 3T_{5}^{2} - 3 \)
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\( T_{7}^{3} + 9T_{7}^{2} + 24T_{7} + 19 \)
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\( T_{13}^{3} - 9T_{13}^{2} + 24T_{13} - 19 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} - 3T^{2} + 3 \)
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|
| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} + 3T^{2} - 3 \)
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|
| $7$ |
\( T^{3} + 9 T^{2} + \cdots + 19 \)
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|
| $11$ |
\( T^{3} + 9 T^{2} + \cdots - 9 \)
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|
| $13$ |
\( T^{3} - 9 T^{2} + \cdots - 19 \)
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| $17$ |
\( T^{3} + 3 T^{2} + \cdots - 111 \)
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|
| $19$ |
\( T^{3} \)
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| $23$ |
\( T^{3} - 3T^{2} + 3 \)
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|
| $29$ |
\( T^{3} - 6 T^{2} + \cdots + 51 \)
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| $31$ |
\( T^{3} - 57T + 107 \)
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| $37$ |
\( T^{3} - 27 T^{2} + \cdots - 613 \)
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| $41$ |
\( T^{3} - 21 T^{2} + \cdots - 321 \)
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| $43$ |
\( T^{3} + 18 T^{2} + \cdots + 127 \)
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| $47$ |
\( T^{3} + 6 T^{2} + \cdots - 51 \)
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|
| $53$ |
\( T^{3} - 6 T^{2} + \cdots + 51 \)
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| $59$ |
\( T^{3} - 3 T^{2} + \cdots + 57 \)
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| $61$ |
\( T^{3} + 9 T^{2} + \cdots - 71 \)
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| $67$ |
\( T^{3} - 9 T^{2} + \cdots - 19 \)
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| $71$ |
\( T^{3} - 3 T^{2} + \cdots + 57 \)
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| $73$ |
\( T^{3} + 18 T^{2} + \cdots - 107 \)
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| $79$ |
\( T^{3} - 201T + 1007 \)
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| $83$ |
\( T^{3} - 27T + 27 \)
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| $89$ |
\( T^{3} - 33 T^{2} + \cdots - 381 \)
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| $97$ |
\( T^{3} - 3 T^{2} + \cdots + 17 \)
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