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gp:[N,k,chi] = [34,4,Mod(1,34)]
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("34.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,4,6,8,-4]
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 \(\beta = \sqrt{13}\).
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
|
| \(2\) |
\( -1 \) |
| \(17\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(34))\):
\( T_{3}^{2} - 6T_{3} - 4 \)
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|
\( T_{5}^{2} + 4T_{5} - 204 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 2)^{2} \)
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| $3$ |
\( T^{2} - 6T - 4 \)
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|
| $5$ |
\( T^{2} + 4T - 204 \)
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|
| $7$ |
\( T^{2} + 6T - 4 \)
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|
| $11$ |
\( T^{2} + 6T - 2916 \)
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| $13$ |
\( T^{2} + 64T - 1524 \)
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|
| $17$ |
\( (T - 17)^{2} \)
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| $19$ |
\( T^{2} + 36T - 2224 \)
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| $23$ |
\( T^{2} - 42T - 612 \)
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| $29$ |
\( T^{2} - 428T + 40596 \)
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| $31$ |
\( T^{2} - 86T - 5028 \)
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| $37$ |
\( T^{2} - 340T + 21412 \)
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| $41$ |
\( T^{2} - 404T - 59868 \)
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| $43$ |
\( T^{2} + 620T + 91888 \)
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| $47$ |
\( T^{2} + 56T - 283968 \)
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| $53$ |
\( T^{2} - 28T - 300156 \)
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| $59$ |
\( T^{2} - 276T + 7344 \)
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| $61$ |
\( T^{2} + 236T - 16028 \)
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| $67$ |
\( T^{2} - 536T - 141168 \)
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| $71$ |
\( T^{2} - 1542 T + 574668 \)
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| $73$ |
\( T^{2} + 164T - 309644 \)
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| $79$ |
\( T^{2} + 1854 T + 704876 \)
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| $83$ |
\( T^{2} + 372T - 22032 \)
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| $89$ |
\( T^{2} + 1976 T + 974844 \)
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| $97$ |
\( T^{2} + 220T - 71100 \)
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