Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,15,Mod(5,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.5");
S:= CuspForms(chi, 15);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.d (of order \(4\), degree \(2\), 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: | \(16.1627658597\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −179.362 | + | 179.362i | −287.384 | − | 47957.1i | 65262.4 | − | 65262.4i | 51545.7 | − | 51545.7i | −714231. | − | 714231.i | 5.66300e6 | + | 5.66300e6i | −4.70038e6 | 2.34111e7i | |||||||
5.2 | −136.719 | + | 136.719i | −534.953 | − | 21000.1i | −103498. | + | 103498.i | 73138.1 | − | 73138.1i | −203533. | − | 203533.i | 631109. | + | 631109.i | −4.49679e6 | − | 2.83004e7i | ||||||
5.3 | −133.495 | + | 133.495i | 4038.24 | − | 19257.7i | −11146.6 | + | 11146.6i | −539083. | + | 539083.i | 192538. | + | 192538.i | 383617. | + | 383617.i | 1.15244e7 | − | 2.97602e6i | ||||||
5.4 | −126.421 | + | 126.421i | −3278.37 | − | 15580.6i | 3422.71 | − | 3422.71i | 414455. | − | 414455.i | 1.01518e6 | + | 1.01518e6i | −101567. | − | 101567.i | 5.96475e6 | 865406.i | |||||||
5.5 | −96.2318 | + | 96.2318i | 907.445 | − | 2137.13i | 41176.3 | − | 41176.3i | −87325.1 | + | 87325.1i | 180476. | + | 180476.i | −1.37100e6 | − | 1.37100e6i | −3.95951e6 | 7.92494e6i | |||||||
5.6 | −58.6742 | + | 58.6742i | −2939.90 | 9498.68i | 36290.0 | − | 36290.0i | 172496. | − | 172496.i | −1.04136e6 | − | 1.04136e6i | −1.51865e6 | − | 1.51865e6i | 3.86002e6 | 4.25857e6i | ||||||||
5.7 | −20.7406 | + | 20.7406i | 2104.15 | 15523.7i | −63748.0 | + | 63748.0i | −43641.4 | + | 43641.4i | −524901. | − | 524901.i | −661784. | − | 661784.i | −355511. | − | 2.64434e6i | |||||||
5.8 | −2.45889 | + | 2.45889i | 917.714 | 16371.9i | 29189.6 | − | 29189.6i | −2256.56 | + | 2256.56i | 560447. | + | 560447.i | −80543.3 | − | 80543.3i | −3.94077e6 | 143548.i | ||||||||
5.9 | 16.8258 | − | 16.8258i | −2685.14 | 15817.8i | −66936.2 | + | 66936.2i | −45179.7 | + | 45179.7i | 261049. | + | 261049.i | 541821. | + | 541821.i | 2.42703e6 | 2.25251e6i | ||||||||
5.10 | 54.4963 | − | 54.4963i | 3813.48 | 10444.3i | 89674.7 | − | 89674.7i | 207821. | − | 207821.i | −863917. | − | 863917.i | 1.46204e6 | + | 1.46204e6i | 9.75965e6 | − | 9.77389e6i | |||||||
5.11 | 80.0104 | − | 80.0104i | −2425.32 | 3580.67i | 98501.3 | − | 98501.3i | −194051. | + | 194051.i | 242708. | + | 242708.i | 1.59738e6 | + | 1.59738e6i | 1.09922e6 | − | 1.57623e7i | |||||||
5.12 | 98.9834 | − | 98.9834i | 3262.77 | − | 3211.43i | −67810.1 | + | 67810.1i | 322960. | − | 322960.i | 947177. | + | 947177.i | 1.30387e6 | + | 1.30387e6i | 5.86269e6 | 1.34241e7i | |||||||
5.13 | 100.811 | − | 100.811i | −1095.41 | − | 3941.67i | −6132.14 | + | 6132.14i | −110429. | + | 110429.i | −84903.1 | − | 84903.1i | 1.25432e6 | + | 1.25432e6i | −3.58306e6 | 1.23637e6i | |||||||
5.14 | 130.772 | − | 130.772i | 566.857 | − | 17818.4i | −36294.1 | + | 36294.1i | 74128.8 | − | 74128.8i | −877300. | − | 877300.i | −187580. | − | 187580.i | −4.46164e6 | 9.49246e6i | |||||||
5.15 | 166.640 | − | 166.640i | 1755.93 | − | 39153.9i | 51367.0 | − | 51367.0i | 292608. | − | 292608.i | 487210. | + | 487210.i | −3.79437e6 | − | 3.79437e6i | −1.69970e6 | − | 1.71196e7i | ||||||
5.16 | 168.563 | − | 168.563i | −4122.10 | − | 40443.0i | −39164.6 | + | 39164.6i | −694834. | + | 694834.i | 252271. | + | 252271.i | −4.05546e6 | − | 4.05546e6i | 1.22088e7 | 1.32034e7i | |||||||
8.1 | −179.362 | − | 179.362i | −287.384 | 47957.1i | 65262.4 | + | 65262.4i | 51545.7 | + | 51545.7i | −714231. | + | 714231.i | 5.66300e6 | − | 5.66300e6i | −4.70038e6 | − | 2.34111e7i | |||||||
8.2 | −136.719 | − | 136.719i | −534.953 | 21000.1i | −103498. | − | 103498.i | 73138.1 | + | 73138.1i | −203533. | + | 203533.i | 631109. | − | 631109.i | −4.49679e6 | 2.83004e7i | ||||||||
8.3 | −133.495 | − | 133.495i | 4038.24 | 19257.7i | −11146.6 | − | 11146.6i | −539083. | − | 539083.i | 192538. | − | 192538.i | 383617. | − | 383617.i | 1.15244e7 | 2.97602e6i | ||||||||
8.4 | −126.421 | − | 126.421i | −3278.37 | 15580.6i | 3422.71 | + | 3422.71i | 414455. | + | 414455.i | 1.01518e6 | − | 1.01518e6i | −101567. | + | 101567.i | 5.96475e6 | − | 865406.i | |||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.15.d.a | ✓ | 32 |
13.d | odd | 4 | 1 | inner | 13.15.d.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.15.d.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
13.15.d.a | ✓ | 32 | 13.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(13, [\chi])\).