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gp:[N,k,chi] = [2475,4,Mod(1,2475)]
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("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [3,-4,0,22,0,0,4,-48,0,0,-33]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
| \(11\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
\( T_{2}^{3} + 4T_{2}^{2} - 15T_{2} - 44 \)
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\( T_{7}^{3} - 4T_{7}^{2} - 428T_{7} + 3744 \)
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\( T_{29}^{3} + 68T_{29}^{2} - 54364T_{29} - 3163056 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} + 4 T^{2} + \cdots - 44 \)
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|
| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 3744 \)
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|
| $11$ |
\( (T + 11)^{3} \)
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|
| $13$ |
\( T^{3} - 3560T + 34144 \)
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| $17$ |
\( T^{3} + 218 T^{2} + \cdots - 235104 \)
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| $19$ |
\( T^{3} - 146 T^{2} + \cdots - 30960 \)
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| $23$ |
\( T^{3} + 200 T^{2} + \cdots + 1664 \)
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| $29$ |
\( T^{3} + 68 T^{2} + \cdots - 3163056 \)
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| $31$ |
\( T^{3} + 68 T^{2} + \cdots - 1812096 \)
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| $37$ |
\( T^{3} - 390 T^{2} + \cdots - 618952 \)
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| $41$ |
\( T^{3} - 196 T^{2} + \cdots + 4364208 \)
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| $43$ |
\( T^{3} - 524 T^{2} + \cdots + 31273920 \)
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| $47$ |
\( T^{3} + 60 T^{2} + \cdots - 20966976 \)
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| $53$ |
\( T^{3} + 158 T^{2} + \cdots + 39574952 \)
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| $59$ |
\( T^{3} - 1044 T^{2} + \cdots + 84227264 \)
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| $61$ |
\( T^{3} - 642 T^{2} + \cdots + 22757384 \)
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| $67$ |
\( T^{3} - 236 T^{2} + \cdots - 87537664 \)
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| $71$ |
\( T^{3} - 544 T^{2} + \cdots - 6553600 \)
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| $73$ |
\( T^{3} + 900 T^{2} + \cdots + 5609344 \)
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| $79$ |
\( T^{3} + 1586 T^{2} + \cdots - 14694992 \)
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| $83$ |
\( T^{3} + 1582 T^{2} + \cdots - 924645384 \)
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| $89$ |
\( T^{3} - 2122 T^{2} + \cdots - 293444632 \)
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| $97$ |
\( T^{3} + 618 T^{2} + \cdots + 223543736 \)
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