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gp:[N,k,chi] = [21,8,Mod(1,21)]
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("21.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,12]
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 \(\beta = 2\sqrt{67}\).
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
|
| \(3\) |
\( +1 \) |
| \(7\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 12T_{2} - 232 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 12T - 232 \)
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|
| $3$ |
\( (T + 27)^{2} \)
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|
| $5$ |
\( T^{2} + 24T - 17008 \)
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|
| $7$ |
\( (T + 343)^{2} \)
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|
| $11$ |
\( T^{2} - 2124 T - 19883356 \)
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|
| $13$ |
\( T^{2} + 1084 T - 21935228 \)
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|
| $17$ |
\( T^{2} + 29256 T - 287563248 \)
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|
| $19$ |
\( T^{2} + 25816 T - 160026224 \)
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|
| $23$ |
\( T^{2} + \cdots + 1166614596 \)
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|
| $29$ |
\( T^{2} + \cdots + 1307845988 \)
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|
| $31$ |
\( T^{2} + \cdots + 43635483648 \)
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|
| $37$ |
\( T^{2} + 28428 T - 887182812 \)
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|
| $41$ |
\( T^{2} + \cdots + 127701459200 \)
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|
| $43$ |
\( T^{2} + \cdots - 71585337968 \)
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|
| $47$ |
\( T^{2} + \cdots - 268603868016 \)
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|
| $53$ |
\( T^{2} + \cdots - 263627226684 \)
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|
| $59$ |
\( T^{2} + \cdots - 345691376064 \)
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|
| $61$ |
\( T^{2} + \cdots + 504531767044 \)
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|
| $67$ |
\( T^{2} + \cdots - 11591287019456 \)
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|
| $71$ |
\( T^{2} + \cdots - 3929864540796 \)
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|
| $73$ |
\( T^{2} + \cdots - 10787980935996 \)
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|
| $79$ |
\( T^{2} + \cdots - 26777501165936 \)
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|
| $83$ |
\( T^{2} + \cdots - 7065815171184 \)
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|
| $89$ |
\( T^{2} + \cdots - 6900412319488 \)
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|
| $97$ |
\( T^{2} + \cdots + 164115180472068 \)
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