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gp:[N,k,chi] = [90,11,Mod(37,90)]
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("90.37");
S:= CuspForms(chi, 11);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 11, names="a")
Newform invariants
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sage:traces = [2,-32,0,0,-5850]
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 \(i = \sqrt{-1}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) |
\(11\) |
\(37\) |
| \(\chi(n)\) |
\(1\) |
\(i\) |
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)
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(90, [\chi])\):
\( T_{7}^{2} - 13906T_{7} + 96688418 \)
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\( T_{11} + 75242 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 32T + 512 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} + 5850 T + 9765625 \)
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|
| $7$ |
\( T^{2} - 13906 T + 96688418 \)
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|
| $11$ |
\( (T + 75242)^{2} \)
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|
| $13$ |
\( T^{2} + \cdots + 24137120898 \)
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|
| $17$ |
\( T^{2} + \cdots + 4675235542658 \)
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|
| $19$ |
\( T^{2} + 16310936142400 \)
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|
| $23$ |
\( T^{2} + \cdots + 1015093061858 \)
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| $29$ |
\( T^{2} + 199023054400 \)
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| $31$ |
\( (T + 29080718)^{2} \)
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| $37$ |
\( T^{2} + \cdots + 1662929902818 \)
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| $41$ |
\( (T - 163945678)^{2} \)
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| $43$ |
\( T^{2} + \cdots + 28\!\cdots\!58 \)
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| $47$ |
\( T^{2} + \cdots + 15\!\cdots\!38 \)
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| $53$ |
\( T^{2} + \cdots + 19\!\cdots\!18 \)
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| $59$ |
\( T^{2} + 88\!\cdots\!00 \)
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| $61$ |
\( (T + 1353610038)^{2} \)
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| $67$ |
\( T^{2} + \cdots + 14\!\cdots\!38 \)
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| $71$ |
\( (T + 2827014562)^{2} \)
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| $73$ |
\( T^{2} + \cdots + 15\!\cdots\!78 \)
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| $79$ |
\( T^{2} + 11\!\cdots\!00 \)
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| $83$ |
\( T^{2} + \cdots + 36\!\cdots\!18 \)
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| $89$ |
\( T^{2} + 71\!\cdots\!00 \)
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| $97$ |
\( T^{2} + \cdots + 55\!\cdots\!18 \)
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