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gp:[N,k,chi] = [882,6,Mod(1,882)]
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("882.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, 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,-8,0,32,-18,0,0,-128,0,72,-2]
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 \(\beta = \sqrt{4705}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(882))\):
\( T_{5}^{2} + 18T_{5} - 4624 \)
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|
\( T_{11}^{2} + 2T_{11} - 381104 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} + 18T - 4624 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 2T - 381104 \)
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|
| $13$ |
\( T^{2} - 288T - 54544 \)
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|
| $17$ |
\( T^{2} + 1530 T - 473400 \)
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|
| $19$ |
\( T^{2} - 1188 T - 117664 \)
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|
| $23$ |
\( T^{2} + 3390 T - 6654600 \)
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| $29$ |
\( T^{2} - 3976 T - 9767636 \)
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| $31$ |
\( T^{2} - 7596 T + 4469024 \)
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| $37$ |
\( T^{2} - 2688 T - 11913444 \)
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| $41$ |
\( T^{2} + 36630 T + 328282920 \)
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| $43$ |
\( T^{2} + 23032 T + 35055376 \)
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| $47$ |
\( T^{2} + 864 T - 412046656 \)
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| $53$ |
\( T^{2} - 32920 T + 147453580 \)
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| $59$ |
\( T^{2} - 26712 T - 732505264 \)
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| $61$ |
\( T^{2} - 20412 T - 459101344 \)
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| $67$ |
\( T^{2} + 36172 T - 479314784 \)
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| $71$ |
\( T^{2} + 73706 T - 252025016 \)
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| $73$ |
\( T^{2} + 74772 T - 152471584 \)
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| $79$ |
\( T^{2} + 23116 T + 95476864 \)
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| $83$ |
\( T^{2} + \cdots + 5438001744 \)
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|
| $89$ |
\( T^{2} + \cdots + 6710448824 \)
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| $97$ |
\( T^{2} + \cdots + 3353807744 \)
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