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gp:[N,k,chi] = [414,6,Mod(1,414)]
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("414.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,-8,0,32,68]
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 \(\beta = 2\sqrt{2}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(23\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 68T_{5} - 3844 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(414))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 68T - 3844 \)
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|
| $7$ |
\( T^{2} + 312T + 22024 \)
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|
| $11$ |
\( T^{2} - 12T - 488036 \)
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|
| $13$ |
\( T^{2} + 1482 T + 548569 \)
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|
| $17$ |
\( T^{2} + 1768 T - 137912 \)
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|
| $19$ |
\( T^{2} + 1124T - 17084 \)
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|
| $23$ |
\( (T + 529)^{2} \)
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|
| $29$ |
\( T^{2} - 5118 T - 23996447 \)
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|
| $31$ |
\( T^{2} + 3110 T - 2069983 \)
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|
| $37$ |
\( T^{2} - 36 T - 166713476 \)
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|
| $41$ |
\( T^{2} - 7234 T + 8654401 \)
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|
| $43$ |
\( T^{2} - 28816 T + 184252352 \)
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|
| $47$ |
\( T^{2} - 19918 T + 96499753 \)
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|
| $53$ |
\( T^{2} - 8508 T - 506524316 \)
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|
| $59$ |
\( T^{2} - 46944 T + 548298976 \)
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|
| $61$ |
\( T^{2} + 61860 T + 551109700 \)
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|
| $67$ |
\( T^{2} + 25428 T - 312058404 \)
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|
| $71$ |
\( T^{2} + \cdots - 2668148783 \)
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|
| $73$ |
\( T^{2} - 11610 T - 392388407 \)
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|
| $79$ |
\( T^{2} + 11340 T - 943610588 \)
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|
| $83$ |
\( T^{2} + \cdots - 2420534536 \)
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|
| $89$ |
\( T^{2} + \cdots + 3214870000 \)
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|
| $97$ |
\( T^{2} + \cdots + 1114766072 \)
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