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gp:[N,k,chi] = [13,8,Mod(1,13)]
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("13.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{337})\).
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
|
| \(13\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 19T_{2} + 6 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 19T + 6 \)
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|
| $3$ |
\( T^{2} - 45T - 252 \)
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|
| $5$ |
\( T^{2} + 353T + 20958 \)
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|
| $7$ |
\( T^{2} + 2009 T + 1002196 \)
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|
| $11$ |
\( T^{2} + 1810 T - 31775952 \)
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|
| $13$ |
\( (T - 2197)^{2} \)
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|
| $17$ |
\( T^{2} + 25361 T - 177216678 \)
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|
| $19$ |
\( T^{2} + \cdots - 1646636688 \)
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|
| $23$ |
\( T^{2} + \cdots - 3712002048 \)
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|
| $29$ |
\( T^{2} + \cdots - 1049753004 \)
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|
| $31$ |
\( T^{2} - 39744 T - 634808464 \)
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|
| $37$ |
\( T^{2} + \cdots - 7868862106 \)
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|
| $41$ |
\( T^{2} + \cdots - 115487893152 \)
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|
| $43$ |
\( T^{2} + \cdots + 11546069484 \)
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|
| $47$ |
\( T^{2} + \cdots + 669689975052 \)
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|
| $53$ |
\( T^{2} + \cdots - 648240166944 \)
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|
| $59$ |
\( T^{2} + \cdots - 140057008272 \)
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|
| $61$ |
\( T^{2} + \cdots + 4947441275696 \)
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|
| $67$ |
\( T^{2} + \cdots - 5405517426256 \)
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|
| $71$ |
\( T^{2} + \cdots - 24787522246284 \)
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|
| $73$ |
\( T^{2} + \cdots + 10156859198164 \)
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|
| $79$ |
\( T^{2} + \cdots - 22472459585984 \)
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|
| $83$ |
\( T^{2} + \cdots + 34361915476704 \)
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|
| $89$ |
\( T^{2} + \cdots - 10393898367132 \)
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|
| $97$ |
\( T^{2} + \cdots + 45398949212204 \)
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