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gp:[N,k,chi] = [528,6,Mod(1,528)]
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("528.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(528, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,0,18,0,-30,0,-48]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = \sqrt{409}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(11\) |
\( -1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):
\( T_{5}^{2} + 30T_{5} - 3456 \)
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|
\( T_{7}^{2} + 48T_{7} - 1060 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( T^{2} + 30T - 3456 \)
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|
| $7$ |
\( T^{2} + 48T - 1060 \)
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|
| $11$ |
\( (T - 121)^{2} \)
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|
| $13$ |
\( T^{2} - 414T - 255312 \)
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|
| $17$ |
\( T^{2} + 1174 T - 411672 \)
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|
| $19$ |
\( T^{2} + 2694 T + 1814000 \)
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|
| $23$ |
\( T^{2} + 2274 T - 5942032 \)
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| $29$ |
\( T^{2} - 1282 T + 263232 \)
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| $31$ |
\( T^{2} - 1720 T + 733056 \)
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|
| $37$ |
\( T^{2} + 2856 T - 50379892 \)
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| $41$ |
\( T^{2} + 7198 T - 32400800 \)
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| $43$ |
\( T^{2} - 7274 T + 8368440 \)
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| $47$ |
\( T^{2} - 3362 T - 39844800 \)
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| $53$ |
\( T^{2} + 44138 T + 438359536 \)
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| $59$ |
\( T^{2} - 27668 T - 663805088 \)
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| $61$ |
\( T^{2} + 25050 T + 48398600 \)
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|
| $67$ |
\( T^{2} + 29080 T + 153588816 \)
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|
| $71$ |
\( T^{2} + \cdots - 1373294400 \)
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|
| $73$ |
\( T^{2} + 52784 T + 548314428 \)
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| $79$ |
\( T^{2} + \cdots + 1811113492 \)
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|
| $83$ |
\( T^{2} - 15196 T - 75154496 \)
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|
| $89$ |
\( T^{2} + \cdots + 12219365460 \)
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| $97$ |
\( T^{2} + \cdots - 1711401532 \)
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