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gp:[N,k,chi] = [110,4,Mod(1,110)]
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("110.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, 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,-4,7,8,-10]
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\).
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
|
| \(2\) |
\( +1 \) |
| \(5\) |
\( +1 \) |
| \(11\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(110))\):
\( T_{3}^{2} - 7T_{3} - 32 \)
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\( T_{7}^{2} - 27T_{7} + 138 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 2)^{2} \)
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|
| $3$ |
\( T^{2} - 7T - 32 \)
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|
| $5$ |
\( (T + 5)^{2} \)
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|
| $7$ |
\( T^{2} - 27T + 138 \)
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|
| $11$ |
\( (T + 11)^{2} \)
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|
| $13$ |
\( T^{2} - 42T - 3984 \)
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|
| $17$ |
\( T^{2} - 49T - 2984 \)
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|
| $19$ |
\( T^{2} - 133T + 4024 \)
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|
| $23$ |
\( T^{2} - 162T + 6384 \)
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| $29$ |
\( T^{2} + 407T + 33934 \)
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| $31$ |
\( T^{2} - 351T + 28632 \)
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| $37$ |
\( T^{2} + 9T - 15954 \)
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| $41$ |
\( T^{2} - 40T - 5972 \)
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| $43$ |
\( T^{2} + 514T + 26224 \)
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| $47$ |
\( T^{2} - 98T - 167696 \)
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| $53$ |
\( T^{2} + 53T - 334226 \)
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| $59$ |
\( T^{2} + 986T + 234376 \)
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|
| $61$ |
\( T^{2} + 25T - 669302 \)
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| $67$ |
\( T^{2} - 172T - 10304 \)
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| $71$ |
\( T^{2} + 427T - 119072 \)
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| $73$ |
\( T^{2} - 646T - 222944 \)
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| $79$ |
\( T^{2} + 706T + 123016 \)
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| $83$ |
\( T^{2} + 492T - 961836 \)
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| $89$ |
\( T^{2} + 273T - 13626 \)
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| $97$ |
\( T^{2} - 3222 T + 2531424 \)
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