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gp:[N,k,chi] = [40,8,Mod(1,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("40.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,0,76]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 \(\beta = 2\sqrt{601}\).
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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 76T_{3} - 960 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(40))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} - 76T - 960 \)
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|
| $5$ |
\( (T - 125)^{2} \)
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|
| $7$ |
\( T^{2} - 796T + 40608 \)
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|
| $11$ |
\( T^{2} + 560 T - 33394896 \)
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|
| $13$ |
\( T^{2} - 4476 T - 36878652 \)
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|
| $17$ |
\( T^{2} - 29524 T + 204031140 \)
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|
| $19$ |
\( T^{2} + \cdots - 1337097136 \)
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|
| $23$ |
\( T^{2} + \cdots + 2607106848 \)
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|
| $29$ |
\( T^{2} + \cdots + 1288412196 \)
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|
| $31$ |
\( T^{2} - 165352 T + 818772480 \)
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|
| $37$ |
\( T^{2} + \cdots - 89047076604 \)
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|
| $41$ |
\( T^{2} + \cdots + 144020798148 \)
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|
| $43$ |
\( T^{2} + \cdots - 43299367968 \)
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|
| $47$ |
\( T^{2} + \cdots - 5081205376 \)
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|
| $53$ |
\( T^{2} + \cdots - 146124685596 \)
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|
| $59$ |
\( T^{2} + \cdots - 3116336362992 \)
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|
| $61$ |
\( T^{2} + \cdots - 1481676069660 \)
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|
| $67$ |
\( T^{2} + \cdots + 1993275644768 \)
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|
| $71$ |
\( T^{2} + \cdots + 24088912922880 \)
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|
| $73$ |
\( T^{2} + \cdots - 20652866457852 \)
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|
| $79$ |
\( T^{2} + \cdots - 12567463439360 \)
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|
| $83$ |
\( T^{2} + \cdots - 10686074681088 \)
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|
| $89$ |
\( T^{2} + \cdots + 8019315835684 \)
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|
| $97$ |
\( T^{2} + \cdots + 64901904650404 \)
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