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gp:[N,k,chi] = [3,20,Mod(1,3)]
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("3.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 20, 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 \(\beta = 3\sqrt{87481}\).
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
|
| \(3\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 702T_{2} - 664128 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 702T - 664128 \)
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|
| $3$ |
\( (T + 19683)^{2} \)
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|
| $5$ |
\( T^{2} + \cdots - 4576715617500 \)
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|
| $7$ |
\( T^{2} + \cdots + 24397550813440 \)
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|
| $11$ |
\( T^{2} + \cdots - 84\!\cdots\!28 \)
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|
| $13$ |
\( T^{2} + \cdots - 77\!\cdots\!08 \)
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|
| $17$ |
\( T^{2} + \cdots - 23\!\cdots\!24 \)
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|
| $19$ |
\( T^{2} + \cdots - 75\!\cdots\!20 \)
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|
| $23$ |
\( T^{2} + \cdots - 13\!\cdots\!80 \)
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|
| $29$ |
\( T^{2} + \cdots + 19\!\cdots\!80 \)
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|
| $31$ |
\( T^{2} + \cdots + 91\!\cdots\!00 \)
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|
| $37$ |
\( T^{2} + \cdots - 97\!\cdots\!80 \)
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|
| $41$ |
\( T^{2} + \cdots - 99\!\cdots\!60 \)
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| $43$ |
\( T^{2} + \cdots + 83\!\cdots\!76 \)
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| $47$ |
\( T^{2} + \cdots - 97\!\cdots\!76 \)
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| $53$ |
\( T^{2} + \cdots + 17\!\cdots\!80 \)
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|
| $59$ |
\( T^{2} + \cdots + 76\!\cdots\!40 \)
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|
| $61$ |
\( T^{2} + \cdots + 18\!\cdots\!16 \)
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| $67$ |
\( T^{2} + \cdots + 60\!\cdots\!76 \)
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| $71$ |
\( T^{2} + \cdots - 75\!\cdots\!16 \)
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| $73$ |
\( T^{2} + \cdots - 23\!\cdots\!00 \)
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| $79$ |
\( T^{2} + \cdots - 12\!\cdots\!00 \)
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| $83$ |
\( T^{2} + \cdots - 53\!\cdots\!12 \)
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| $89$ |
\( T^{2} + \cdots - 82\!\cdots\!00 \)
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| $97$ |
\( T^{2} + \cdots + 10\!\cdots\!16 \)
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