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gp:[N,k,chi] = [231,4,Mod(1,231)]
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("231.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,-3,6,-3,-19]
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\).
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
|
| \(3\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
| \(11\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(231))\):
\( T_{2}^{2} + 3T_{2} - 2 \)
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\( T_{5}^{2} + 19T_{5} + 52 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 3T - 2 \)
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|
| $3$ |
\( (T - 3)^{2} \)
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|
| $5$ |
\( T^{2} + 19T + 52 \)
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|
| $7$ |
\( (T - 7)^{2} \)
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|
| $11$ |
\( (T - 11)^{2} \)
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|
| $13$ |
\( T^{2} - 11T - 2218 \)
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|
| $17$ |
\( T^{2} + 146T + 5312 \)
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| $19$ |
\( T^{2} + 101T - 1534 \)
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|
| $23$ |
\( T^{2} + 184T + 7376 \)
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| $29$ |
\( T^{2} + 217T + 10238 \)
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| $31$ |
\( T^{2} - 20T - 4252 \)
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| $37$ |
\( T^{2} + 173T - 54742 \)
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| $41$ |
\( T^{2} + 60T - 82400 \)
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| $43$ |
\( T^{2} + 164T + 6112 \)
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| $47$ |
\( T^{2} + 467T - 64006 \)
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| $53$ |
\( T^{2} - 326T + 3296 \)
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| $59$ |
\( T^{2} + 781T + 109136 \)
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| $61$ |
\( T^{2} + 148T - 64156 \)
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| $67$ |
\( T^{2} - 473T - 62596 \)
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| $71$ |
\( T^{2} + 756T - 470816 \)
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| $73$ |
\( T^{2} + 213T + 10624 \)
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| $79$ |
\( T^{2} - 168 T - 1005056 \)
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| $83$ |
\( T^{2} + 2096 T + 1068316 \)
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| $89$ |
\( T^{2} - 736 T - 1333988 \)
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| $97$ |
\( T^{2} - 638T - 265592 \)
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