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gp:[N,k,chi] = [735,6,Mod(1,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("735.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-4,18]
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 \(\beta = 4\sqrt{2}\).
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 intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(735))\):
\( T_{2}^{2} + 4T_{2} - 28 \)
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|
\( T_{13}^{2} + 692T_{13} + 114308 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 4T - 28 \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( (T + 25)^{2} \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 176T - 312256 \)
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|
| $13$ |
\( T^{2} + 692T + 114308 \)
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|
| $17$ |
\( T^{2} - 428 T - 1603132 \)
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|
| $19$ |
\( T^{2} - 1824 T + 724096 \)
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|
| $23$ |
\( T^{2} - 8032 T + 15079904 \)
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|
| $29$ |
\( T^{2} + 2948 T - 8333852 \)
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|
| $31$ |
\( T^{2} + 14760 T + 51973488 \)
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|
| $37$ |
\( T^{2} - 156 T - 176563548 \)
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|
| $41$ |
\( T^{2} - 5980 T - 85728700 \)
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| $43$ |
\( T^{2} - 25672 T + 124406768 \)
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| $47$ |
\( T^{2} + 21904 T + 112481056 \)
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| $53$ |
\( T^{2} + 53948 T + 726392324 \)
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|
| $59$ |
\( T^{2} + 26296 T - 882188144 \)
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|
| $61$ |
\( T^{2} - 16788 T - 251308476 \)
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|
| $67$ |
\( T^{2} - 16264 T - 4247824 \)
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| $71$ |
\( T^{2} - 22264 T + 123338224 \)
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| $73$ |
\( T^{2} + 28684 T - 157081084 \)
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|
| $79$ |
\( T^{2} + \cdots - 2543446976 \)
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|
| $83$ |
\( T^{2} + \cdots - 1894968176 \)
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|
| $89$ |
\( T^{2} + \cdots - 8699912828 \)
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|
| $97$ |
\( T^{2} + \cdots + 14224705732 \)
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