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gp:[N,k,chi] = [105,6,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, 6);
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, 6, names="a")
Newform invariants
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sage:traces = [2,1]
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 \(\beta = \frac{1}{2}(1 + \sqrt{233})\).
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
\( T_{2}^{2} - T_{2} - 58 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(105))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - T - 58 \)
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|
| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( (T + 25)^{2} \)
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|
| $7$ |
\( (T + 49)^{2} \)
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|
| $11$ |
\( T^{2} + 196T - 13696 \)
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|
| $13$ |
\( T^{2} - 32T - 157252 \)
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|
| $17$ |
\( T^{2} - 932 T - 2302972 \)
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|
| $19$ |
\( T^{2} + 1164T + 69376 \)
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|
| $23$ |
\( T^{2} - 1832 T - 238336 \)
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|
| $29$ |
\( T^{2} + 1108 T - 1337132 \)
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|
| $31$ |
\( T^{2} + 13740 T + 38062368 \)
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|
| $37$ |
\( T^{2} + 8244 T - 11226348 \)
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|
| $41$ |
\( T^{2} - 2620 T + 224900 \)
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|
| $43$ |
\( T^{2} - 15472 T - 127210432 \)
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|
| $47$ |
\( T^{2} + 9016 T + 18174736 \)
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|
| $53$ |
\( T^{2} - 24752 T + 135158204 \)
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|
| $59$ |
\( T^{2} + 38224 T - 142339664 \)
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|
| $61$ |
\( T^{2} + 14868 T - 419865516 \)
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|
| $67$ |
\( T^{2} - 65464 T + 97924736 \)
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|
| $71$ |
\( T^{2} - 34244 T + 140291584 \)
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|
| $73$ |
\( T^{2} + 59696 T + 863009276 \)
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|
| $79$ |
\( T^{2} + \cdots + 5673522304 \)
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|
| $83$ |
\( T^{2} + \cdots + 1224050224 \)
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|
| $89$ |
\( T^{2} + \cdots + 3058544452 \)
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|
| $97$ |
\( T^{2} + \cdots - 19767762068 \)
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