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gp:[N,k,chi] = [850,4,Mod(1,850)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("850.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
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sage:traces = [3,6,-7,12,0,-14,5]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(17\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(850))\):
\( T_{3}^{3} + 7T_{3}^{2} - 7T_{3} - 67 \)
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\( T_{7}^{3} - 5T_{7}^{2} - 645T_{7} - 3921 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 2)^{3} \)
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|
| $3$ |
\( T^{3} + 7 T^{2} + \cdots - 67 \)
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|
| $5$ |
\( T^{3} \)
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|
| $7$ |
\( T^{3} - 5 T^{2} + \cdots - 3921 \)
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|
| $11$ |
\( T^{3} + 6 T^{2} + \cdots - 8100 \)
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|
| $13$ |
\( T^{3} + 39 T^{2} + \cdots - 14135 \)
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|
| $17$ |
\( (T - 17)^{3} \)
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|
| $19$ |
\( T^{3} + 110 T^{2} + \cdots - 378296 \)
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|
| $23$ |
\( T^{3} + 206 T^{2} + \cdots + 279204 \)
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|
| $29$ |
\( T^{3} - 12 T^{2} + \cdots + 441936 \)
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| $31$ |
\( T^{3} + 125 T^{2} + \cdots - 69053 \)
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| $37$ |
\( T^{3} + 52 T^{2} + \cdots + 1247184 \)
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| $41$ |
\( T^{3} + 242 T^{2} + \cdots - 36269208 \)
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| $43$ |
\( T^{3} + 656 T^{2} + \cdots - 7992272 \)
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| $47$ |
\( T^{3} + 138 T^{2} + \cdots - 1034856 \)
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| $53$ |
\( T^{3} + 1175 T^{2} + \cdots + 16379397 \)
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| $59$ |
\( T^{3} - 468 T^{2} + \cdots + 56883312 \)
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| $61$ |
\( T^{3} + 1158 T^{2} + \cdots - 2065016 \)
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| $67$ |
\( T^{3} + 372 T^{2} + \cdots + 46770016 \)
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| $71$ |
\( T^{3} + 485 T^{2} + \cdots - 689439 \)
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| $73$ |
\( T^{3} + 412 T^{2} + \cdots + 5708208 \)
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| $79$ |
\( T^{3} - 613 T^{2} + \cdots + 405833083 \)
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| $83$ |
\( T^{3} + 652 T^{2} + \cdots - 50888640 \)
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| $89$ |
\( T^{3} - 1018 T^{2} + \cdots + 9591384 \)
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| $97$ |
\( T^{3} + 1956 T^{2} + \cdots - 92513600 \)
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