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gp:[N,k,chi] = [210,4,Mod(1,210)]
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("210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,4,6,8,10,12,14]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = 4\sqrt{106}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(210))\):
\( T_{11}^{2} - 32T_{11} - 1440 \)
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\( T_{13}^{2} + 12T_{13} - 6748 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 2)^{2} \)
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| $3$ |
\( (T - 3)^{2} \)
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|
| $5$ |
\( (T - 5)^{2} \)
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|
| $7$ |
\( (T - 7)^{2} \)
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|
| $11$ |
\( T^{2} - 32T - 1440 \)
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|
| $13$ |
\( T^{2} + 12T - 6748 \)
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|
| $17$ |
\( T^{2} - 4T - 1692 \)
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|
| $19$ |
\( T^{2} - 64T - 14240 \)
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|
| $23$ |
\( T^{2} - 32T - 1440 \)
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|
| $29$ |
\( T^{2} + 84T - 13500 \)
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| $31$ |
\( T^{2} + 72T - 5488 \)
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| $37$ |
\( T^{2} - 60T - 796 \)
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| $41$ |
\( T^{2} + 92T - 106428 \)
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| $43$ |
\( T^{2} + 184T + 6768 \)
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| $47$ |
\( T^{2} + 208T - 31584 \)
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| $53$ |
\( T^{2} + 300T + 7236 \)
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| $59$ |
\( T^{2} + 568T + 53520 \)
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| $61$ |
\( T^{2} + 516T + 24164 \)
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| $67$ |
\( T^{2} - 392T - 3984 \)
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| $71$ |
\( T^{2} + 1448 T + 497040 \)
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| $73$ |
\( T^{2} + 756T - 26716 \)
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| $79$ |
\( T^{2} - 416T + 36480 \)
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| $83$ |
\( T^{2} + 1752 T + 523152 \)
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| $89$ |
\( T^{2} + 956T + 119940 \)
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| $97$ |
\( T^{2} - 684 T - 1029532 \)
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