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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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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);
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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")
Newform invariants
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sage:traces = [2,3,0,181,330,0,0,1185,0,4820,-2844,0,-2534]
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 \(\beta = \frac{1}{2}(1 + \sqrt{865})\).
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 \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
\( T_{2}^{2} - 3T_{2} - 214 \)
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\( T_{5}^{2} - 330T_{5} + 5600 \)
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|
\( T_{13}^{2} + 2534T_{13} - 166620776 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 3T - 214 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 330T + 5600 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 2844 T - 887776 \)
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|
| $13$ |
\( T^{2} + 2534 T - 166620776 \)
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|
| $17$ |
\( T^{2} + 1488 T - 22147524 \)
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|
| $19$ |
\( T^{2} + 32810 T + 109928560 \)
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|
| $23$ |
\( T^{2} - 6576 T + 10312704 \)
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|
| $29$ |
\( T^{2} + \cdots - 18920124100 \)
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|
| $31$ |
\( T^{2} + \cdots + 37023636384 \)
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|
| $37$ |
\( T^{2} + \cdots - 126010986084 \)
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| $41$ |
\( T^{2} + \cdots + 13303276364 \)
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| $43$ |
\( T^{2} + \cdots + 50864711104 \)
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| $47$ |
\( T^{2} + \cdots + 2090896416 \)
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| $53$ |
\( T^{2} + \cdots + 2037435782724 \)
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|
| $59$ |
\( T^{2} + \cdots - 615374101440 \)
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|
| $61$ |
\( T^{2} + \cdots - 529516501136 \)
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| $67$ |
\( T^{2} + \cdots - 533876854064 \)
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| $71$ |
\( T^{2} + \cdots + 10359492378624 \)
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| $73$ |
\( T^{2} + \cdots - 3340687254156 \)
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| $79$ |
\( T^{2} + \cdots - 6335206025600 \)
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| $83$ |
\( T^{2} + \cdots + 30181573873584 \)
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|
| $89$ |
\( T^{2} + \cdots - 4649674734460 \)
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| $97$ |
\( T^{2} + \cdots + 27021168617436 \)
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