Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3696,2,Mod(769,3696)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3696.769");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3696.q (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: | \(29.5127085871\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 1848) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
769.1 | 0 | − | 1.00000i | 0 | − | 3.54741i | 0 | −1.12129 | + | 2.39640i | 0 | −1.00000 | 0 | ||||||||||||||
769.2 | 0 | − | 1.00000i | 0 | − | 2.64261i | 0 | −1.38114 | − | 2.25664i | 0 | −1.00000 | 0 | ||||||||||||||
769.3 | 0 | − | 1.00000i | 0 | − | 1.71581i | 0 | −1.06269 | + | 2.42295i | 0 | −1.00000 | 0 | ||||||||||||||
769.4 | 0 | − | 1.00000i | 0 | − | 1.64449i | 0 | −2.63155 | − | 0.273742i | 0 | −1.00000 | 0 | ||||||||||||||
769.5 | 0 | − | 1.00000i | 0 | − | 1.09147i | 0 | 2.61768 | + | 0.384362i | 0 | −1.00000 | 0 | ||||||||||||||
769.6 | 0 | − | 1.00000i | 0 | − | 0.331450i | 0 | 1.23509 | − | 2.33977i | 0 | −1.00000 | 0 | ||||||||||||||
769.7 | 0 | − | 1.00000i | 0 | 0.210416i | 0 | 2.16457 | + | 1.52140i | 0 | −1.00000 | 0 | |||||||||||||||
769.8 | 0 | − | 1.00000i | 0 | 0.883273i | 0 | 0.763345 | − | 2.53324i | 0 | −1.00000 | 0 | |||||||||||||||
769.9 | 0 | − | 1.00000i | 0 | 2.32410i | 0 | 0.212212 | + | 2.63723i | 0 | −1.00000 | 0 | |||||||||||||||
769.10 | 0 | − | 1.00000i | 0 | 2.50231i | 0 | −2.45571 | − | 0.984620i | 0 | −1.00000 | 0 | |||||||||||||||
769.11 | 0 | − | 1.00000i | 0 | 3.27939i | 0 | −2.61990 | + | 0.368974i | 0 | −1.00000 | 0 | |||||||||||||||
769.12 | 0 | − | 1.00000i | 0 | 3.77374i | 0 | 2.27938 | − | 1.34328i | 0 | −1.00000 | 0 | |||||||||||||||
769.13 | 0 | 1.00000i | 0 | − | 3.77374i | 0 | 2.27938 | + | 1.34328i | 0 | −1.00000 | 0 | |||||||||||||||
769.14 | 0 | 1.00000i | 0 | − | 3.27939i | 0 | −2.61990 | − | 0.368974i | 0 | −1.00000 | 0 | |||||||||||||||
769.15 | 0 | 1.00000i | 0 | − | 2.50231i | 0 | −2.45571 | + | 0.984620i | 0 | −1.00000 | 0 | |||||||||||||||
769.16 | 0 | 1.00000i | 0 | − | 2.32410i | 0 | 0.212212 | − | 2.63723i | 0 | −1.00000 | 0 | |||||||||||||||
769.17 | 0 | 1.00000i | 0 | − | 0.883273i | 0 | 0.763345 | + | 2.53324i | 0 | −1.00000 | 0 | |||||||||||||||
769.18 | 0 | 1.00000i | 0 | − | 0.210416i | 0 | 2.16457 | − | 1.52140i | 0 | −1.00000 | 0 | |||||||||||||||
769.19 | 0 | 1.00000i | 0 | 0.331450i | 0 | 1.23509 | + | 2.33977i | 0 | −1.00000 | 0 | ||||||||||||||||
769.20 | 0 | 1.00000i | 0 | 1.09147i | 0 | 2.61768 | − | 0.384362i | 0 | −1.00000 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3696.2.q.f | 24 | |
4.b | odd | 2 | 1 | 1848.2.q.b | yes | 24 | |
7.b | odd | 2 | 1 | 3696.2.q.g | 24 | ||
11.b | odd | 2 | 1 | 3696.2.q.g | 24 | ||
28.d | even | 2 | 1 | 1848.2.q.a | ✓ | 24 | |
44.c | even | 2 | 1 | 1848.2.q.a | ✓ | 24 | |
77.b | even | 2 | 1 | inner | 3696.2.q.f | 24 | |
308.g | odd | 2 | 1 | 1848.2.q.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1848.2.q.a | ✓ | 24 | 28.d | even | 2 | 1 | |
1848.2.q.a | ✓ | 24 | 44.c | even | 2 | 1 | |
1848.2.q.b | yes | 24 | 4.b | odd | 2 | 1 | |
1848.2.q.b | yes | 24 | 308.g | odd | 2 | 1 | |
3696.2.q.f | 24 | 1.a | even | 1 | 1 | trivial | |
3696.2.q.f | 24 | 77.b | even | 2 | 1 | inner | |
3696.2.q.g | 24 | 7.b | odd | 2 | 1 | ||
3696.2.q.g | 24 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3696, [\chi])\):
\( T_{5}^{24} + 64 T_{5}^{22} + 1742 T_{5}^{20} + 26412 T_{5}^{18} + 245601 T_{5}^{16} + 1454284 T_{5}^{14} + \cdots + 16384 \) |
\( T_{13}^{12} - 92 T_{13}^{10} - 32 T_{13}^{9} + 2921 T_{13}^{8} + 1384 T_{13}^{7} - 37978 T_{13}^{6} + \cdots + 241664 \) |