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gp:[N,k,chi] = [225,8,Mod(1,225)]
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("225.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,-8,0,24,0,0,86]
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 \(\beta = 2\sqrt{31}\).
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
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 \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):
\( T_{2}^{2} + 8T_{2} - 108 \)
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|
\( T_{7}^{2} - 86T_{7} - 22455 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 8T - 108 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 86T - 22455 \)
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|
| $11$ |
\( T^{2} - 4612 T - 3456108 \)
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|
| $13$ |
\( T^{2} - 9018 T + 9466697 \)
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|
| $17$ |
\( T^{2} - 19948 T - 506147724 \)
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|
| $19$ |
\( T^{2} + 23934 T - 791814895 \)
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|
| $23$ |
\( T^{2} + \cdots - 1770541740 \)
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|
| $29$ |
\( T^{2} + \cdots + 29988616500 \)
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|
| $31$ |
\( T^{2} + \cdots - 25626949575 \)
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|
| $37$ |
\( T^{2} + \cdots - 205209213980 \)
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|
| $41$ |
\( T^{2} + \cdots - 186336929520 \)
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| $43$ |
\( T^{2} + \cdots + 285507233321 \)
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| $47$ |
\( T^{2} + \cdots + 19283408964 \)
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| $53$ |
\( T^{2} + \cdots - 1554251453280 \)
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|
| $59$ |
\( T^{2} + \cdots - 1475873866620 \)
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|
| $61$ |
\( T^{2} + \cdots - 8423736487399 \)
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|
| $67$ |
\( T^{2} + \cdots + 5094434196801 \)
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| $71$ |
\( T^{2} + \cdots - 284467184496 \)
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| $73$ |
\( T^{2} + \cdots + 3729052906820 \)
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| $79$ |
\( T^{2} + \cdots + 9582807148800 \)
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|
| $83$ |
\( T^{2} + \cdots - 12878345203452 \)
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|
| $89$ |
\( T^{2} + \cdots + 45203910693120 \)
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|
| $97$ |
\( T^{2} + \cdots + 42934761529441 \)
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