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gp:[N,k,chi] = [350,6,Mod(1,350)]
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("350.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, 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,-8,20]
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 \(\beta = 2\sqrt{79}\).
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
|
| \(2\) |
\( +1 \) |
| \(5\) |
\( -1 \) |
| \(7\) |
\( +1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):
\( T_{3}^{2} - 20T_{3} - 216 \)
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|
\( T_{13}^{2} - 736T_{13} - 130332 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} - 20T - 216 \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( (T + 49)^{2} \)
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|
| $11$ |
\( T^{2} + 1070 T + 283381 \)
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|
| $13$ |
\( T^{2} - 736T - 130332 \)
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|
| $17$ |
\( T^{2} + 1904 T + 116304 \)
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|
| $19$ |
\( T^{2} - 828 T - 3935340 \)
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|
| $23$ |
\( T^{2} - 1654 T + 251325 \)
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|
| $29$ |
\( T^{2} + 2246 T + 937545 \)
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|
| $31$ |
\( T^{2} - 288 T - 57595228 \)
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|
| $37$ |
\( T^{2} - 8986 T - 27876551 \)
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|
| $41$ |
\( T^{2} - 11228 T + 3642952 \)
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| $43$ |
\( T^{2} - 3402 T - 183005923 \)
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| $47$ |
\( T^{2} - 15572 T + 56935972 \)
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|
| $53$ |
\( T^{2} + 16188 T - 220879228 \)
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|
| $59$ |
\( T^{2} + \cdots + 2124282020 \)
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|
| $61$ |
\( T^{2} - 15172 T - 28227960 \)
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|
| $67$ |
\( T^{2} - 78966 T + 682884189 \)
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|
| $71$ |
\( T^{2} + \cdots - 1567614723 \)
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| $73$ |
\( T^{2} + \cdots + 1233469320 \)
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| $79$ |
\( T^{2} + 72106 T + 131273245 \)
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| $83$ |
\( T^{2} + \cdots - 6215296876 \)
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|
| $89$ |
\( T^{2} + \cdots - 1086335900 \)
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|
| $97$ |
\( T^{2} + \cdots - 10668157580 \)
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