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gp:[N,k,chi] = [171,4,Mod(1,171)]
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("171.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [3,3,0,9,12]
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
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(171))\):
\( T_{2}^{3} - 3T_{2}^{2} - 12T_{2} - 6 \)
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|
\( T_{5}^{3} - 12T_{5}^{2} - 147T_{5} - 294 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} - 3 T^{2} + \cdots - 6 \)
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|
| $3$ |
\( T^{3} \)
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|
| $5$ |
\( T^{3} - 12 T^{2} + \cdots - 294 \)
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|
| $7$ |
\( T^{3} + 18 T^{2} + \cdots + 1096 \)
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|
| $11$ |
\( T^{3} - 54 T^{2} + \cdots + 123768 \)
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|
| $13$ |
\( T^{3} + 66 T^{2} + \cdots - 118952 \)
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|
| $17$ |
\( T^{3} - 48 T^{2} + \cdots - 197646 \)
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|
| $19$ |
\( (T - 19)^{3} \)
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| $23$ |
\( T^{3} - 456 T^{2} + \cdots - 2984448 \)
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| $29$ |
\( T^{3} + 30 T^{2} + \cdots + 322440 \)
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| $31$ |
\( T^{3} + 144 T^{2} + \cdots - 4391408 \)
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| $37$ |
\( T^{3} - 162 T^{2} + \cdots - 1184384 \)
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| $41$ |
\( T^{3} - 354 T^{2} + \cdots - 2020992 \)
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| $43$ |
\( T^{3} + 246 T^{2} + \cdots - 1958012 \)
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| $47$ |
\( T^{3} - 258 T^{2} + \cdots + 4021584 \)
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| $53$ |
\( T^{3} - 906 T^{2} + \cdots + 123717192 \)
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| $59$ |
\( T^{3} + 420 T^{2} + \cdots - 3603840 \)
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| $61$ |
\( T^{3} - 996 T^{2} + \cdots + 2953882 \)
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| $67$ |
\( T^{3} - 492 T^{2} + \cdots + 28256896 \)
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| $71$ |
\( T^{3} - 624 T^{2} + \cdots + 13790208 \)
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| $73$ |
\( T^{3} - 564 T^{2} + \cdots + 52873318 \)
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| $79$ |
\( T^{3} + 300 T^{2} + \cdots - 67955840 \)
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| $83$ |
\( T^{3} + 504 T^{2} + \cdots - 10222848 \)
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| $89$ |
\( T^{3} + 930 T^{2} + \cdots - 482046600 \)
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| $97$ |
\( T^{3} + 618 T^{2} + \cdots + 35556376 \)
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