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gp:[N,k,chi] = [441,8,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,9,0,317,-360,0,0,5067,0,9030,4932,0,-7708]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{1065})\).
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
|
| \(3\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
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 \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
\( T_{2}^{2} - 9T_{2} - 246 \)
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\( T_{5}^{2} + 360T_{5} - 74100 \)
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|
\( T_{13}^{2} + 7708T_{13} - 7230524 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 9T - 246 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} + 360T - 74100 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 4932 T - 19176384 \)
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|
| $13$ |
\( T^{2} + 7708 T - 7230524 \)
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|
| $17$ |
\( T^{2} + 28584 T + 121953804 \)
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|
| $19$ |
\( T^{2} - 63728 T + 782053936 \)
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|
| $23$ |
\( T^{2} + \cdots + 1392518400 \)
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|
| $29$ |
\( T^{2} + \cdots + 46362346164 \)
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|
| $31$ |
\( T^{2} + \cdots - 27226807040 \)
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|
| $37$ |
\( T^{2} + \cdots + 123432685924 \)
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| $41$ |
\( T^{2} + \cdots - 416101387716 \)
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| $43$ |
\( T^{2} + \cdots + 23523686224 \)
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| $47$ |
\( T^{2} + \cdots + 551244428160 \)
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| $53$ |
\( T^{2} + \cdots + 800144528964 \)
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| $59$ |
\( T^{2} + \cdots - 455709488016 \)
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|
| $61$ |
\( T^{2} + \cdots - 397808236556 \)
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| $67$ |
\( T^{2} + \cdots + 4656119933104 \)
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| $71$ |
\( T^{2} + \cdots - 387209643840 \)
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| $73$ |
\( T^{2} + \cdots - 22405484001836 \)
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| $79$ |
\( T^{2} + \cdots + 578840156416 \)
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| $83$ |
\( T^{2} + \cdots + 6817674434256 \)
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|
| $89$ |
\( T^{2} + \cdots - 61835691772164 \)
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| $97$ |
\( T^{2} + \cdots + 62174212264276 \)
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