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gp:[N,k,chi] = [1134,2,Mod(215,1134)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 5]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1134.215");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [4,0,0,-4,0,0,2,0,0,-6,0]
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
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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 a primitive root of unity \(\zeta_{12}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).
| \(n\) |
\(325\) |
\(407\) |
| \(\chi(n)\) |
\(1 - \zeta_{12}^{2}\) |
\(1 - \zeta_{12}^{2}\) |
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)
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):
\( T_{5}^{4} + 3T_{5}^{2} + 9 \)
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\( T_{11}^{4} - 9T_{11}^{2} + 81 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + 3T^{2} + 9 \)
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| $7$ |
\( (T^{2} - T + 7)^{2} \)
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| $11$ |
\( T^{4} - 9T^{2} + 81 \)
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| $13$ |
\( (T^{2} - 6 T + 12)^{2} \)
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| $17$ |
\( T^{4} + 12T^{2} + 144 \)
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| $19$ |
\( (T^{2} - 6 T + 12)^{2} \)
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| $23$ |
\( T^{4} - 36T^{2} + 1296 \)
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| $29$ |
\( T^{4} - 9T^{2} + 81 \)
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| $31$ |
\( (T^{2} + 3)^{2} \)
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| $37$ |
\( (T^{2} - 2 T + 4)^{2} \)
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| $41$ |
\( T^{4} + 48T^{2} + 2304 \)
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| $43$ |
\( (T^{2} - 8 T + 64)^{2} \)
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| $47$ |
\( (T^{2} - 48)^{2} \)
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| $53$ |
\( T^{4} - 81T^{2} + 6561 \)
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| $59$ |
\( (T^{2} - 3)^{2} \)
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| $61$ |
\( T^{4} \)
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| $67$ |
\( (T - 2)^{4} \)
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| $71$ |
\( (T^{2} + 144)^{2} \)
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| $73$ |
\( (T^{2} - 12 T + 48)^{2} \)
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| $79$ |
\( (T + 1)^{4} \)
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| $83$ |
\( T^{4} + 75T^{2} + 5625 \)
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| $89$ |
\( T^{4} + 108 T^{2} + 11664 \)
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| $97$ |
\( (T^{2} + 9 T + 27)^{2} \)
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