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gp:[N,k,chi] = [75,6,Mod(1,75)]
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("75.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, 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 = [2,6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = \sqrt{31}\).
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 kernel of the linear operator
\( T_{2}^{2} - 6T_{2} - 22 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 6T - 22 \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 102T - 15255 \)
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|
| $11$ |
\( T^{2} + 12T - 240028 \)
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|
| $13$ |
\( T^{2} - 1054T - 7967 \)
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|
| $17$ |
\( T^{2} + 1716 T + 686564 \)
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|
| $19$ |
\( T^{2} - 4214 T + 3564505 \)
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|
| $23$ |
\( T^{2} + 444 T - 12967740 \)
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|
| $29$ |
\( T^{2} - 4068 T + 4039940 \)
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|
| $31$ |
\( T^{2} + 2598 T - 25471575 \)
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|
| $37$ |
\( T^{2} - 4412 T - 92913020 \)
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|
| $41$ |
\( T^{2} - 11232 T + 28823360 \)
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|
| $43$ |
\( T^{2} + 8450 T + 16404289 \)
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|
| $47$ |
\( T^{2} - 2460 T - 341830204 \)
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|
| $53$ |
\( T^{2} + \cdots + 1056424400 \)
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|
| $59$ |
\( T^{2} + 63924 T + 741079460 \)
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|
| $61$ |
\( T^{2} - 7310 T - 18138959 \)
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|
| $67$ |
\( T^{2} - 61734 T + 127378089 \)
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|
| $71$ |
\( T^{2} + \cdots + 1224279104 \)
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|
| $73$ |
\( T^{2} + 26564 T - 183636860 \)
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|
| $79$ |
\( T^{2} + \cdots + 1649721600 \)
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|
| $83$ |
\( T^{2} + \cdots + 1063204452 \)
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|
| $89$ |
\( T^{2} + \cdots + 2346485760 \)
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|
| $97$ |
\( T^{2} + \cdots - 13910130431 \)
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