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gp:[N,k,chi] = [576,6,Mod(1,576)]
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("576.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,0,0,0,36,0,120,0,0,0,200]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 \(\beta = 16\sqrt{31}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(576))\):
\( T_{5}^{2} - 36T_{5} - 7612 \)
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\( T_{7}^{2} - 120T_{7} - 4336 \)
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\( T_{11}^{2} - 200T_{11} - 275696 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} - 36T - 7612 \)
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|
| $7$ |
\( T^{2} - 120T - 4336 \)
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|
| $11$ |
\( T^{2} - 200T - 275696 \)
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|
| $13$ |
\( T^{2} + 284T - 265532 \)
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|
| $17$ |
\( T^{2} + 2676 T + 1504548 \)
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|
| $19$ |
\( T^{2} - 72T - 792304 \)
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| $23$ |
\( T^{2} - 3840 T - 3456000 \)
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| $29$ |
\( T^{2} - 10212 T + 23206340 \)
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| $31$ |
\( T^{2} + 10488 T + 27301136 \)
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| $37$ |
\( T^{2} + 13148 T - 12778940 \)
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| $41$ |
\( T^{2} + 4164 T - 15505276 \)
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| $43$ |
\( T^{2} + 5832 T - 125615344 \)
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| $47$ |
\( T^{2} - 1520 T - 310545344 \)
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| $53$ |
\( T^{2} - 9012 T - 19701340 \)
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| $59$ |
\( T^{2} - 55096 T + 388630288 \)
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| $61$ |
\( T^{2} - 63444 T + 988000740 \)
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| $67$ |
\( T^{2} + \cdots - 1551497968 \)
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| $71$ |
\( T^{2} + \cdots - 1167829760 \)
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| $73$ |
\( T^{2} + \cdots - 2060240220 \)
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| $79$ |
\( T^{2} + \cdots + 4557502736 \)
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| $83$ |
\( T^{2} + \cdots + 2593974672 \)
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| $89$ |
\( T^{2} + \cdots - 2331558940 \)
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| $97$ |
\( T^{2} + \cdots + 1164856516 \)
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