Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1280,3,Mod(639,1280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1280.639");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1280 = 2^{8} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1280.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(34.8774738381\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 640) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
639.1 | 0 | − | 4.06912i | 0 | 4.37877 | + | 2.41379i | 0 | −13.1670 | 0 | −7.55776 | 0 | |||||||||||||||
639.2 | 0 | 4.06912i | 0 | 4.37877 | − | 2.41379i | 0 | −13.1670 | 0 | −7.55776 | 0 | ||||||||||||||||
639.3 | 0 | − | 1.73799i | 0 | −3.35996 | − | 3.70279i | 0 | −7.32638 | 0 | 5.97938 | 0 | |||||||||||||||
639.4 | 0 | 1.73799i | 0 | −3.35996 | + | 3.70279i | 0 | −7.32638 | 0 | 5.97938 | 0 | ||||||||||||||||
639.5 | 0 | − | 0.933412i | 0 | 0.723593 | + | 4.94736i | 0 | 6.91082 | 0 | 8.12874 | 0 | |||||||||||||||
639.6 | 0 | 0.933412i | 0 | 0.723593 | − | 4.94736i | 0 | 6.91082 | 0 | 8.12874 | 0 | ||||||||||||||||
639.7 | 0 | − | 5.46407i | 0 | 3.70957 | − | 3.35247i | 0 | −5.17181 | 0 | −20.8560 | 0 | |||||||||||||||
639.8 | 0 | 5.46407i | 0 | 3.70957 | + | 3.35247i | 0 | −5.17181 | 0 | −20.8560 | 0 | ||||||||||||||||
639.9 | 0 | − | 4.20423i | 0 | −1.52118 | − | 4.76298i | 0 | 3.29905 | 0 | −8.67551 | 0 | |||||||||||||||
639.10 | 0 | 4.20423i | 0 | −1.52118 | + | 4.76298i | 0 | 3.29905 | 0 | −8.67551 | 0 | ||||||||||||||||
639.11 | 0 | − | 2.00470i | 0 | 4.99385 | − | 0.247851i | 0 | 4.85433 | 0 | 4.98119 | 0 | |||||||||||||||
639.12 | 0 | 2.00470i | 0 | 4.99385 | + | 0.247851i | 0 | 4.85433 | 0 | 4.98119 | 0 | ||||||||||||||||
639.13 | 0 | − | 2.00470i | 0 | −4.99385 | + | 0.247851i | 0 | −4.85433 | 0 | 4.98119 | 0 | |||||||||||||||
639.14 | 0 | 2.00470i | 0 | −4.99385 | − | 0.247851i | 0 | −4.85433 | 0 | 4.98119 | 0 | ||||||||||||||||
639.15 | 0 | − | 4.20423i | 0 | 1.52118 | + | 4.76298i | 0 | −3.29905 | 0 | −8.67551 | 0 | |||||||||||||||
639.16 | 0 | 4.20423i | 0 | 1.52118 | − | 4.76298i | 0 | −3.29905 | 0 | −8.67551 | 0 | ||||||||||||||||
639.17 | 0 | − | 5.46407i | 0 | −3.70957 | + | 3.35247i | 0 | 5.17181 | 0 | −20.8560 | 0 | |||||||||||||||
639.18 | 0 | 5.46407i | 0 | −3.70957 | − | 3.35247i | 0 | 5.17181 | 0 | −20.8560 | 0 | ||||||||||||||||
639.19 | 0 | − | 0.933412i | 0 | −0.723593 | − | 4.94736i | 0 | −6.91082 | 0 | 8.12874 | 0 | |||||||||||||||
639.20 | 0 | 0.933412i | 0 | −0.723593 | + | 4.94736i | 0 | −6.91082 | 0 | 8.12874 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1280.3.e.k | 24 | |
4.b | odd | 2 | 1 | 1280.3.e.l | 24 | ||
5.b | even | 2 | 1 | inner | 1280.3.e.k | 24 | |
8.b | even | 2 | 1 | 1280.3.e.l | 24 | ||
8.d | odd | 2 | 1 | inner | 1280.3.e.k | 24 | |
16.e | even | 4 | 1 | 640.3.h.a | ✓ | 24 | |
16.e | even | 4 | 1 | 640.3.h.b | yes | 24 | |
16.f | odd | 4 | 1 | 640.3.h.a | ✓ | 24 | |
16.f | odd | 4 | 1 | 640.3.h.b | yes | 24 | |
20.d | odd | 2 | 1 | 1280.3.e.l | 24 | ||
40.e | odd | 2 | 1 | inner | 1280.3.e.k | 24 | |
40.f | even | 2 | 1 | 1280.3.e.l | 24 | ||
80.k | odd | 4 | 1 | 640.3.h.a | ✓ | 24 | |
80.k | odd | 4 | 1 | 640.3.h.b | yes | 24 | |
80.q | even | 4 | 1 | 640.3.h.a | ✓ | 24 | |
80.q | even | 4 | 1 | 640.3.h.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
640.3.h.a | ✓ | 24 | 16.e | even | 4 | 1 | |
640.3.h.a | ✓ | 24 | 16.f | odd | 4 | 1 | |
640.3.h.a | ✓ | 24 | 80.k | odd | 4 | 1 | |
640.3.h.a | ✓ | 24 | 80.q | even | 4 | 1 | |
640.3.h.b | yes | 24 | 16.e | even | 4 | 1 | |
640.3.h.b | yes | 24 | 16.f | odd | 4 | 1 | |
640.3.h.b | yes | 24 | 80.k | odd | 4 | 1 | |
640.3.h.b | yes | 24 | 80.q | even | 4 | 1 | |
1280.3.e.k | 24 | 1.a | even | 1 | 1 | trivial | |
1280.3.e.k | 24 | 5.b | even | 2 | 1 | inner | |
1280.3.e.k | 24 | 8.d | odd | 2 | 1 | inner | |
1280.3.e.k | 24 | 40.e | odd | 2 | 1 | inner | |
1280.3.e.l | 24 | 4.b | odd | 2 | 1 | ||
1280.3.e.l | 24 | 8.b | even | 2 | 1 | ||
1280.3.e.l | 24 | 20.d | odd | 2 | 1 | ||
1280.3.e.l | 24 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\):
\( T_{3}^{12} + 72T_{3}^{10} + 1840T_{3}^{8} + 20320T_{3}^{6} + 93824T_{3}^{4} + 173568T_{3}^{2} + 92416 \) |
\( T_{7}^{12} - 336T_{7}^{10} + 38144T_{7}^{8} - 2008032T_{7}^{6} + 52816128T_{7}^{4} - 661713920T_{7}^{2} + 3048806656 \) |
\( T_{11}^{6} + 4T_{11}^{5} - 348T_{11}^{4} - 1184T_{11}^{3} + 29168T_{11}^{2} + 87872T_{11} - 4672 \) |
\( T_{13}^{12} - 1216 T_{13}^{10} + 564864 T_{13}^{8} - 126646784 T_{13}^{6} + 14349455360 T_{13}^{4} - 771603464192 T_{13}^{2} + 15343327707136 \) |