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gp:[N,k,chi] = [315,6,Mod(1,315)]
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("315.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, 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,-1,0,-31]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\).
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
\( T_{2}^{2} + T_{2} - 16 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(315))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + T - 16 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 25)^{2} \)
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|
| $7$ |
\( (T + 49)^{2} \)
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|
| $11$ |
\( T^{2} - 601T - 62596 \)
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|
| $13$ |
\( T^{2} + 577T + 37586 \)
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|
| $17$ |
\( T^{2} + 41T - 1023346 \)
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|
| $19$ |
\( T^{2} - 630T - 20960 \)
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|
| $23$ |
\( T^{2} - 442 T - 13172224 \)
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|
| $29$ |
\( T^{2} + 5885 T - 5853350 \)
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|
| $31$ |
\( T^{2} + 396 T - 13834656 \)
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|
| $37$ |
\( T^{2} + 8904 T - 5978196 \)
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|
| $41$ |
\( T^{2} + 1774 T - 236091496 \)
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|
| $43$ |
\( T^{2} + 27122 T + 170086856 \)
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|
| $47$ |
\( T^{2} - 21289 T + 72995224 \)
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|
| $53$ |
\( T^{2} - 55582 T + 768990296 \)
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|
| $59$ |
\( T^{2} + 59600 T + 451339840 \)
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|
| $61$ |
\( T^{2} + 51846 T + 142927344 \)
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|
| $67$ |
\( T^{2} + 45344 T + 174936944 \)
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|
| $71$ |
\( T^{2} + \cdots + 1172746624 \)
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|
| $73$ |
\( T^{2} + 13532 T - 239780284 \)
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|
| $79$ |
\( T^{2} + \cdots - 4382959400 \)
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|
| $83$ |
\( T^{2} + \cdots + 2765333536 \)
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|
| $89$ |
\( T^{2} - 37650 T - 177665240 \)
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|
| $97$ |
\( T^{2} + 96339 T - 838817066 \)
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