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gp:[N,k,chi] = [171,6,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, 6);
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, 6, names="a")
Newform invariants
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sage:traces = [2,7,0,49,133]
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\).
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
|
| \(3\) |
\( -1 \) |
| \(19\) |
\( +1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(171))\):
\( T_{2}^{2} - 7T_{2} - 32 \)
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|
\( T_{5}^{2} - 133T_{5} + 3316 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 7T - 32 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 133T + 3316 \)
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|
| $7$ |
\( T^{2} - 72T - 7377 \)
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|
| $11$ |
\( T^{2} - 705T + 116778 \)
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|
| $13$ |
\( T^{2} + 1341 T + 412356 \)
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|
| $17$ |
\( T^{2} + 2784 T + 1859607 \)
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|
| $19$ |
\( (T + 361)^{2} \)
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|
| $23$ |
\( T^{2} - 2713 T - 1457816 \)
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|
| $29$ |
\( T^{2} - 7775 T + 13694842 \)
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|
| $31$ |
\( T^{2} - 7132 T - 21863072 \)
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|
| $37$ |
\( T^{2} + 6248 T - 28020212 \)
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| $41$ |
\( T^{2} - 4174 T - 52950128 \)
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| $43$ |
\( T^{2} - 25357 T + 103337908 \)
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| $47$ |
\( T^{2} + 11727 T + 34377048 \)
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| $53$ |
\( T^{2} - 29133 T + 137515428 \)
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|
| $59$ |
\( T^{2} + \cdots + 1040195802 \)
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|
| $61$ |
\( T^{2} + 40939 T - 280177226 \)
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| $67$ |
\( T^{2} - 19039 T - 80586308 \)
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| $71$ |
\( T^{2} - 70236 T - 170310588 \)
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| $73$ |
\( T^{2} - 67058 T + 555800113 \)
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| $79$ |
\( T^{2} + 32850 T + 93134448 \)
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| $83$ |
\( T^{2} + 71534 T + 666270472 \)
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|
| $89$ |
\( T^{2} + \cdots - 2036009792 \)
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|
| $97$ |
\( T^{2} + 62458 T + 832198864 \)
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