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gp:[N,k,chi] = [98,8,Mod(1,98)]
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("98.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,-16,-56,128,14,448,0,-1024,-908]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == 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 \(\beta = \sqrt{949}\).
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
|
| \(2\) |
\( +1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 56T_{3} - 165 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 8)^{2} \)
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|
| $3$ |
\( T^{2} + 56T - 165 \)
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|
| $5$ |
\( T^{2} - 14T - 185955 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 2408 T - 43237845 \)
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|
| $13$ |
\( T^{2} + 10724 T + 26913780 \)
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|
| $17$ |
\( T^{2} - 35098 T + 198263001 \)
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|
| $19$ |
\( T^{2} - 2408 T - 625809965 \)
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|
| $23$ |
\( T^{2} + 61684 T - 890257137 \)
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|
| $29$ |
\( T^{2} + 95660 T - 302210796 \)
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|
| $31$ |
\( T^{2} + \cdots + 5628386895 \)
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|
| $37$ |
\( T^{2} + \cdots + 46003879093 \)
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|
| $41$ |
\( T^{2} + \cdots + 154982022516 \)
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|
| $43$ |
\( T^{2} + \cdots - 9474621680 \)
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|
| $47$ |
\( T^{2} + \cdots + 776500057791 \)
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|
| $53$ |
\( T^{2} + \cdots + 880940835909 \)
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|
| $59$ |
\( T^{2} + \cdots - 9792650253 \)
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|
| $61$ |
\( T^{2} + \cdots - 1149292144659 \)
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|
| $67$ |
\( T^{2} + \cdots - 3517340463061 \)
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|
| $71$ |
\( T^{2} + \cdots - 20052968986176 \)
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|
| $73$ |
\( T^{2} + \cdots + 4575230600825 \)
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|
| $79$ |
\( T^{2} + \cdots + 12232151046375 \)
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|
| $83$ |
\( T^{2} + \cdots + 14206197364176 \)
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|
| $89$ |
\( T^{2} + \cdots - 61691712832359 \)
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|
| $97$ |
\( T^{2} + \cdots - 89006519008780 \)
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