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gp:[N,k,chi] = [1445,4,Mod(1,1445)]
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("1445.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1445, 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,-4,2,-2,-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 \(\beta = \sqrt{3}\).
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
|
| \(5\) |
\( +1 \) |
| \(17\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):
\( T_{2}^{2} + 4T_{2} + 1 \)
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\( T_{3}^{2} - 2T_{3} - 2 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 4T + 1 \)
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|
| $3$ |
\( T^{2} - 2T - 2 \)
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|
| $5$ |
\( (T + 5)^{2} \)
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|
| $7$ |
\( T^{2} + 2T - 242 \)
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|
| $11$ |
\( T^{2} - 38T - 146 \)
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| $13$ |
\( T^{2} + 80T - 752 \)
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|
| $17$ |
\( T^{2} \)
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| $19$ |
\( T^{2} + 180T + 7992 \)
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|
| $23$ |
\( T^{2} - 42T - 30162 \)
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| $29$ |
\( T^{2} - 216T + 5316 \)
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| $31$ |
\( T^{2} + 70T - 58418 \)
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| $37$ |
\( T^{2} + 352T - 2732 \)
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| $41$ |
\( T^{2} + 344T - 30908 \)
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| $43$ |
\( T^{2} + 88T - 80732 \)
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| $47$ |
\( T^{2} + 348T + 30084 \)
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| $53$ |
\( T^{2} + 164T - 5564 \)
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| $59$ |
\( T^{2} + 1436 T + 502456 \)
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| $61$ |
\( T^{2} - 76T - 214028 \)
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| $67$ |
\( T^{2} + 44T - 284108 \)
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| $71$ |
\( T^{2} - 1510 T + 527542 \)
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| $73$ |
\( T^{2} + 1088 T + 294964 \)
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| $79$ |
\( T^{2} - 958T - 343466 \)
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| $83$ |
\( T^{2} - 248T - 154556 \)
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| $89$ |
\( T^{2} + 204 T - 1641288 \)
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| $97$ |
\( T^{2} + 1012 T - 314252 \)
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