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gp:[N,k,chi] = [1058,6,Mod(1,1058)]
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("1058.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,-8,6,32,68]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\beta = 2\sqrt{55}\).
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
|
| \(2\) |
\( +1 \) |
| \(23\) |
\( -1 \) |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1058))\):
\( T_{3}^{2} - 6T_{3} - 211 \)
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|
\( T_{5}^{2} - 68T_{5} - 4344 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} - 6T - 211 \)
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|
| $5$ |
\( T^{2} - 68T - 4344 \)
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|
| $7$ |
\( T^{2} + 96T + 2084 \)
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|
| $11$ |
\( T^{2} - 684T + 114984 \)
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|
| $13$ |
\( T^{2} + 498T - 25999 \)
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|
| $17$ |
\( T^{2} - 2904 T + 2058804 \)
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|
| $19$ |
\( T^{2} - 2428 T + 1248516 \)
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|
| $23$ |
\( T^{2} \)
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| $29$ |
\( T^{2} - 2226 T - 30195711 \)
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| $31$ |
\( T^{2} - 1222T - 72179 \)
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| $37$ |
\( T^{2} - 5356 T - 6688536 \)
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| $41$ |
\( T^{2} - 886 T - 2563431 \)
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| $43$ |
\( T^{2} + 26720 T + 167841600 \)
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| $47$ |
\( T^{2} + 20594 T + 105948789 \)
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| $53$ |
\( T^{2} + 7412 T - 663099084 \)
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| $59$ |
\( T^{2} - 3600 T - 762790320 \)
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| $61$ |
\( T^{2} + 27132 T + 156777476 \)
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| $67$ |
\( T^{2} + 44300 T + 319867080 \)
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| $71$ |
\( T^{2} + 22962 T - 40496139 \)
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| $73$ |
\( T^{2} + \cdots - 5491841159 \)
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| $79$ |
\( T^{2} + \cdots - 2000697564 \)
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| $83$ |
\( T^{2} + \cdots - 6250229196 \)
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| $89$ |
\( T^{2} + \cdots - 1334898144 \)
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| $97$ |
\( T^{2} + \cdots - 8938195164 \)
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