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gp:[N,k,chi] = [1710,4,Mod(1,1710)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1710.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
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sage:traces = [3,-6,0,12,15,0,-11,-24,0,-30,68]
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(19\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):
\( T_{7}^{3} + 11T_{7}^{2} - 984T_{7} - 16288 \)
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\( T_{11}^{3} - 68T_{11}^{2} - 2812T_{11} + 194816 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 2)^{3} \)
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|
| $3$ |
\( T^{3} \)
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|
| $5$ |
\( (T - 5)^{3} \)
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|
| $7$ |
\( T^{3} + 11 T^{2} + \cdots - 16288 \)
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|
| $11$ |
\( T^{3} - 68 T^{2} + \cdots + 194816 \)
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|
| $13$ |
\( T^{3} + 121 T^{2} + \cdots + 942 \)
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|
| $17$ |
\( T^{3} - 103 T^{2} + \cdots + 1782324 \)
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|
| $19$ |
\( (T + 19)^{3} \)
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|
| $23$ |
\( T^{3} - 203 T^{2} + \cdots + 2255976 \)
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|
| $29$ |
\( T^{3} + 91 T^{2} + \cdots - 61620 \)
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| $31$ |
\( T^{3} + 154 T^{2} + \cdots - 2514272 \)
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| $37$ |
\( T^{3} + 304 T^{2} + \cdots - 3608532 \)
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| $41$ |
\( T^{3} + 282 T^{2} + \cdots + 749864 \)
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| $43$ |
\( T^{3} + 214 T^{2} + \cdots - 14445824 \)
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| $47$ |
\( T^{3} - 1144 T^{2} + \cdots - 51801600 \)
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| $53$ |
\( T^{3} + 795 T^{2} + \cdots - 7559654 \)
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| $59$ |
\( T^{3} - 609 T^{2} + \cdots + 12873880 \)
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| $61$ |
\( T^{3} - 792 T^{2} + \cdots - 6609824 \)
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| $67$ |
\( T^{3} + 865 T^{2} + \cdots - 57516042 \)
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| $71$ |
\( T^{3} - 312 T^{2} + \cdots + 40935424 \)
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| $73$ |
\( T^{3} - 79 T^{2} + \cdots + 356614884 \)
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| $79$ |
\( T^{3} - 1806 T^{2} + \cdots - 89460880 \)
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| $83$ |
\( T^{3} - 34 T^{2} + \cdots + 212872272 \)
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| $89$ |
\( T^{3} + \cdots + 1232043000 \)
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| $97$ |
\( T^{3} + 1120 T^{2} + \cdots + 38577204 \)
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