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gp:[N,k,chi] = [1575,4,Mod(1,1575)]
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("1575.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, 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,-1,0,3,0,0,-21,-21,0,0,66]
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(7\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):
\( T_{2}^{3} + T_{2}^{2} - 13T_{2} - 1 \)
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\( T_{11}^{3} - 66T_{11}^{2} - 276T_{11} + 6120 \)
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\( T_{13}^{3} - 102T_{13}^{2} + 3084T_{13} - 28696 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{3} + T^{2} - 13T - 1 \)
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|
| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} \)
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|
| $7$ |
\( (T + 7)^{3} \)
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|
| $11$ |
\( T^{3} - 66 T^{2} + \cdots + 6120 \)
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| $13$ |
\( T^{3} - 102 T^{2} + \cdots - 28696 \)
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|
| $17$ |
\( T^{3} + 152 T^{2} + \cdots + 68272 \)
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| $19$ |
\( T^{3} + 138 T^{2} + \cdots + 70632 \)
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| $23$ |
\( T^{3} - 180 T^{2} + \cdots + 228816 \)
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| $29$ |
\( T^{3} + 170 T^{2} + \cdots - 1680584 \)
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| $31$ |
\( T^{3} + 366 T^{2} + \cdots - 3510632 \)
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| $37$ |
\( T^{3} - 252 T^{2} + \cdots + 8221888 \)
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| $41$ |
\( T^{3} - 206 T^{2} + \cdots + 18222200 \)
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| $43$ |
\( T^{3} + 108 T^{2} + \cdots + 13701184 \)
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| $47$ |
\( T^{3} - 24 T^{2} + \cdots - 20893824 \)
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| $53$ |
\( T^{3} + 354 T^{2} + \cdots - 53538840 \)
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| $59$ |
\( T^{3} - 880 T^{2} + \cdots + 22878976 \)
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| $61$ |
\( T^{3} + 870 T^{2} + \cdots - 112468792 \)
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| $67$ |
\( T^{3} + 96 T^{2} + \cdots - 35189888 \)
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| $71$ |
\( T^{3} - 1018 T^{2} + \cdots + 133243912 \)
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| $73$ |
\( T^{3} - 1554 T^{2} + \cdots + 199646136 \)
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| $79$ |
\( T^{3} + 1620 T^{2} + \cdots - 258624448 \)
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| $83$ |
\( T^{3} - 872 T^{2} + \cdots + 688370432 \)
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| $89$ |
\( T^{3} - 1938 T^{2} + \cdots - 181473048 \)
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| $97$ |
\( T^{3} - 1878 T^{2} + \cdots + 140508392 \)
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