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gp:[N,k,chi] = [75,6,Mod(1,75)]
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("75.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,-9]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{89})\).
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 9T_{2} - 2 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 9T - 2 \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} + 108T - 9900 \)
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|
| $11$ |
\( T^{2} - 168T - 252468 \)
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|
| $13$ |
\( T^{2} + 1296 T + 262908 \)
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|
| $17$ |
\( T^{2} + 576T - 353156 \)
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|
| $19$ |
\( T^{2} + 1336 T + 330880 \)
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|
| $23$ |
\( T^{2} + 5904 T + 8542000 \)
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|
| $29$ |
\( T^{2} + 3552 T - 13455360 \)
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|
| $31$ |
\( T^{2} + 11648 T + 33457600 \)
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|
| $37$ |
\( T^{2} + 14688 T + 50445180 \)
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|
| $41$ |
\( T^{2} - 1812 T - 202645980 \)
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|
| $43$ |
\( T^{2} - 7560 T - 25902576 \)
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|
| $47$ |
\( T^{2} + 3240 T - 34287104 \)
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|
| $53$ |
\( T^{2} - 22176 T + 32872540 \)
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|
| $59$ |
\( T^{2} - 57336 T + 572451660 \)
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|
| $61$ |
\( T^{2} + 30140 T + 3798916 \)
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|
| $67$ |
\( T^{2} - 5184 T - 455939136 \)
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|
| $71$ |
\( T^{2} - 17424 T - 157672656 \)
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|
| $73$ |
\( T^{2} - 3456 T - 10970640 \)
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|
| $79$ |
\( T^{2} + \cdots - 1122176000 \)
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|
| $83$ |
\( T^{2} + \cdots + 3449918032 \)
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|
| $89$ |
\( T^{2} + \cdots + 4173659460 \)
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|
| $97$ |
\( T^{2} + \cdots - 3585658176 \)
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