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gp:[N,k,chi] = [105,4,Mod(1,105)]
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("105.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, 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]
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 \(\beta = \sqrt{5}\).
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
| \(7\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} - 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 4T - 1 \)
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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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|
| $11$ |
\( T^{2} + 92T + 2096 \)
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|
| $13$ |
\( T^{2} - 8T - 7204 \)
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|
| $17$ |
\( T^{2} + 44T - 9196 \)
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|
| $19$ |
\( T^{2} + 108T - 464 \)
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|
| $23$ |
\( T^{2} + 320T + 23600 \)
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|
| $29$ |
\( T^{2} + 236T + 13204 \)
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|
| $31$ |
\( T^{2} + 60T - 51120 \)
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|
| $37$ |
\( T^{2} - 204T + 7524 \)
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|
| $41$ |
\( T^{2} - 44T - 31516 \)
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|
| $43$ |
\( T^{2} - 136T - 77296 \)
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|
| $47$ |
\( T^{2} - 400T - 101120 \)
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|
| $53$ |
\( T^{2} - 16T - 36916 \)
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|
| $59$ |
\( T^{2} + 464T + 47344 \)
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|
| $61$ |
\( T^{2} + 684T + 81684 \)
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|
| $67$ |
\( T^{2} - 736T + 944 \)
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|
| $71$ |
\( T^{2} + 740T + 41680 \)
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|
| $73$ |
\( T^{2} - 424T - 29476 \)
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|
| $79$ |
\( T^{2} + 408 T - 1129664 \)
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|
| $83$ |
\( T^{2} - 608T + 57136 \)
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|
| $89$ |
\( T^{2} + 1332 T + 380836 \)
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|
| $97$ |
\( T^{2} + 2448 T + 1461196 \)
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