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gp:[N,k,chi] = [90,16,Mod(1,90)]
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("90.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 16, names="a")
Newform invariants
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sage:traces = [2,-256,0,32768,156250,0,-1761200]
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 \(\beta = 18\sqrt{20672521}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( +1 \) |
| \(5\) |
\( -1 \) |
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 \(S_{16}^{\mathrm{new}}(\Gamma_0(90))\):
\( T_{7}^{2} + 1761200T_{7} + 447259416604 \)
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|
\( T_{11}^{2} + 11981640T_{11} - 2597247548387316 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 128)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 78125)^{2} \)
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|
| $7$ |
\( T^{2} + \cdots + 447259416604 \)
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|
| $11$ |
\( T^{2} + \cdots - 25\!\cdots\!16 \)
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|
| $13$ |
\( T^{2} + \cdots + 14\!\cdots\!16 \)
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|
| $17$ |
\( T^{2} + \cdots - 65\!\cdots\!16 \)
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|
| $19$ |
\( T^{2} + \cdots + 21\!\cdots\!16 \)
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|
| $23$ |
\( T^{2} + \cdots - 40\!\cdots\!00 \)
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|
| $29$ |
\( T^{2} + \cdots - 11\!\cdots\!16 \)
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|
| $31$ |
\( T^{2} + \cdots - 53\!\cdots\!84 \)
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| $37$ |
\( T^{2} + \cdots - 21\!\cdots\!96 \)
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| $41$ |
\( T^{2} + \cdots - 11\!\cdots\!00 \)
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| $43$ |
\( T^{2} + \cdots - 66\!\cdots\!76 \)
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| $47$ |
\( T^{2} + \cdots + 93\!\cdots\!00 \)
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| $53$ |
\( T^{2} + \cdots + 50\!\cdots\!04 \)
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| $59$ |
\( T^{2} + \cdots - 75\!\cdots\!16 \)
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| $61$ |
\( T^{2} + \cdots - 87\!\cdots\!56 \)
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| $67$ |
\( T^{2} + \cdots - 22\!\cdots\!64 \)
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| $71$ |
\( T^{2} + \cdots - 26\!\cdots\!04 \)
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| $73$ |
\( T^{2} + \cdots - 24\!\cdots\!00 \)
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| $79$ |
\( T^{2} + \cdots - 10\!\cdots\!64 \)
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| $83$ |
\( T^{2} + \cdots + 29\!\cdots\!76 \)
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| $89$ |
\( T^{2} + \cdots - 18\!\cdots\!84 \)
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| $97$ |
\( T^{2} + \cdots - 19\!\cdots\!16 \)
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