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gp:[N,k,chi] = [2178,4,Mod(1,2178)]
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("2178.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2178, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,4,0,8,6,0,12]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = \sqrt{3}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(11\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):
\( T_{5}^{2} - 6T_{5} - 39 \)
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\( T_{7} - 6 \)
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\( T_{17}^{2} + 60T_{17} + 225 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 2)^{2} \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} - 6T - 39 \)
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| $7$ |
\( (T - 6)^{2} \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} + 60T - 1623 \)
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|
| $17$ |
\( T^{2} + 60T + 225 \)
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| $19$ |
\( T^{2} - 120T + 3408 \)
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|
| $23$ |
\( T^{2} - 48T + 276 \)
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| $29$ |
\( T^{2} - 144T + 2997 \)
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| $31$ |
\( T^{2} - 176T - 44528 \)
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| $37$ |
\( T^{2} - 14T - 8699 \)
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| $41$ |
\( T^{2} - 108T - 30159 \)
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| $43$ |
\( T^{2} - 612T + 50436 \)
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| $47$ |
\( T^{2} - 792T + 146016 \)
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| $53$ |
\( T^{2} - 378T - 104247 \)
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| $59$ |
\( T^{2} - 744T + 126852 \)
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| $61$ |
\( T^{2} - 888T + 131424 \)
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| $67$ |
\( T^{2} - 8T - 13052 \)
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| $71$ |
\( T^{2} - 192 T - 1107084 \)
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| $73$ |
\( T^{2} + 48T - 51696 \)
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| $79$ |
\( T^{2} - 2040 T + 988128 \)
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| $83$ |
\( T^{2} + 672T - 225792 \)
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| $89$ |
\( T^{2} + 1326 T + 128517 \)
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| $97$ |
\( T^{2} - 1666T + 2689 \)
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