Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,10,Mod(5,16)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.5");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.e (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: | \(8.24057337862\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Relative dimension: | \(17\) 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 | −22.5065 | + | 2.33610i | −82.6799 | + | 82.6799i | 501.085 | − | 105.155i | 1214.44 | + | 1214.44i | 1667.69 | − | 2053.98i | − | 16.4286i | −11032.0 | + | 3537.26i | 6011.08i | −30169.9 | − | 24495.7i | |||
5.2 | −21.3919 | − | 7.37468i | 10.9414 | − | 10.9414i | 403.228 | + | 315.517i | −1361.43 | − | 1361.43i | −314.745 | + | 153.367i | − | 1171.64i | −6298.98 | − | 9723.20i | 19443.6i | 19083.5 | + | 39163.8i | |||
5.3 | −20.7667 | + | 8.98584i | 155.975 | − | 155.975i | 350.509 | − | 373.212i | −404.055 | − | 404.055i | −1837.51 | + | 4640.65i | 637.759i | −3925.29 | + | 10900.0i | − | 28973.4i | 12021.7 | + | 4760.11i | |||
5.4 | −16.2245 | − | 15.7723i | 111.916 | − | 111.916i | 14.4713 | + | 511.795i | 1353.12 | + | 1353.12i | −3580.96 | + | 50.6167i | 6223.61i | 7837.38 | − | 8531.89i | − | 5367.52i | −611.981 | − | 43295.6i | |||
5.5 | −15.3981 | + | 16.5801i | −149.427 | + | 149.427i | −37.7985 | − | 510.603i | −1560.48 | − | 1560.48i | −176.625 | − | 4778.42i | 10771.6i | 9047.86 | + | 7235.60i | − | 24974.1i | 49901.2 | − | 1844.50i | |||
5.6 | −12.8269 | + | 18.6406i | −4.54374 | + | 4.54374i | −182.942 | − | 478.201i | 335.590 | + | 335.590i | −26.4158 | − | 142.980i | − | 11538.0i | 11260.5 | + | 2723.70i | 19641.7i | −10560.2 | + | 1951.01i | |||
5.7 | −11.9169 | − | 19.2351i | −128.561 | + | 128.561i | −227.974 | + | 458.445i | −28.4518 | − | 28.4518i | 4004.93 | + | 940.828i | − | 2525.84i | 11535.0 | − | 1078.14i | − | 13372.9i | −208.215 | + | 886.330i | ||
5.8 | −3.01687 | + | 22.4254i | 80.3683 | − | 80.3683i | −493.797 | − | 135.309i | 652.415 | + | 652.415i | 1559.83 | + | 2044.75i | 9250.15i | 4524.08 | − | 10665.4i | 6764.87i | −16598.9 | + | 12662.4i | ||||
5.9 | −0.735037 | − | 22.6155i | 100.059 | − | 100.059i | −510.919 | + | 33.2464i | −601.526 | − | 601.526i | −2336.42 | − | 2189.32i | − | 5438.71i | 1127.43 | + | 11530.2i | − | 340.415i | −13161.7 | + | 14046.0i | ||
5.10 | 7.39503 | + | 21.3849i | −180.934 | + | 180.934i | −402.627 | + | 316.284i | 1444.92 | + | 1444.92i | −5207.26 | − | 2531.24i | − | 567.986i | −9741.14 | − | 6271.20i | − | 45790.9i | −20214.2 | + | 41584.6i | ||
5.11 | 8.52251 | − | 20.9611i | −77.8288 | + | 77.8288i | −366.734 | − | 357.282i | 206.519 | + | 206.519i | 968.080 | + | 2294.67i | 6914.62i | −10614.5 | + | 4642.20i | 7568.35i | 6088.93 | − | 2568.81i | ||||
5.12 | 8.74072 | + | 20.8710i | −8.26401 | + | 8.26401i | −359.200 | + | 364.856i | −1313.02 | − | 1313.02i | −244.712 | − | 100.245i | − | 3840.19i | −10754.6 | − | 4307.77i | 19546.4i | 15927.3 | − | 38880.8i | |||
5.13 | 17.2877 | + | 14.5992i | 185.041 | − | 185.041i | 85.7268 | + | 504.772i | 356.050 | + | 356.050i | 5900.38 | − | 497.477i | − | 6406.17i | −5887.25 | + | 9977.87i | − | 48797.5i | 957.228 | + | 11353.3i | ||
5.14 | 18.5834 | − | 12.9096i | 62.9496 | − | 62.9496i | 178.687 | − | 479.807i | 1829.14 | + | 1829.14i | 357.167 | − | 1982.47i | − | 7941.79i | −2873.49 | − | 11223.2i | 11757.7i | 57605.1 | + | 10378.3i | |||
5.15 | 20.5458 | − | 9.47996i | 117.096 | − | 117.096i | 332.261 | − | 389.547i | −1418.63 | − | 1418.63i | 1295.77 | − | 3515.91i | 7313.79i | 3133.68 | − | 11153.4i | − | 7740.11i | −42595.4 | − | 15698.3i | |||
5.16 | 21.2709 | − | 7.71668i | −159.536 | + | 159.536i | 392.906 | − | 328.282i | −1021.24 | − | 1021.24i | −2162.40 | + | 4624.58i | − | 10191.8i | 5824.23 | − | 10014.8i | − | 31220.8i | −29603.4 | − | 13842.2i | ||
5.17 | 21.4373 | + | 7.24163i | −33.5715 | + | 33.5715i | 407.118 | + | 310.482i | 315.639 | + | 315.639i | −962.796 | + | 476.571i | 3725.04i | 6479.11 | + | 9604.11i | 17428.9i | 4480.71 | + | 9052.18i | ||||
13.1 | −22.5065 | − | 2.33610i | −82.6799 | − | 82.6799i | 501.085 | + | 105.155i | 1214.44 | − | 1214.44i | 1667.69 | + | 2053.98i | 16.4286i | −11032.0 | − | 3537.26i | − | 6011.08i | −30169.9 | + | 24495.7i | |||
13.2 | −21.3919 | + | 7.37468i | 10.9414 | + | 10.9414i | 403.228 | − | 315.517i | −1361.43 | + | 1361.43i | −314.745 | − | 153.367i | 1171.64i | −6298.98 | + | 9723.20i | − | 19443.6i | 19083.5 | − | 39163.8i | |||
13.3 | −20.7667 | − | 8.98584i | 155.975 | + | 155.975i | 350.509 | + | 373.212i | −404.055 | + | 404.055i | −1837.51 | − | 4640.65i | − | 637.759i | −3925.29 | − | 10900.0i | 28973.4i | 12021.7 | − | 4760.11i | |||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 16.10.e.a | ✓ | 34 |
4.b | odd | 2 | 1 | 64.10.e.a | 34 | ||
8.b | even | 2 | 1 | 128.10.e.b | 34 | ||
8.d | odd | 2 | 1 | 128.10.e.a | 34 | ||
16.e | even | 4 | 1 | inner | 16.10.e.a | ✓ | 34 |
16.e | even | 4 | 1 | 128.10.e.b | 34 | ||
16.f | odd | 4 | 1 | 64.10.e.a | 34 | ||
16.f | odd | 4 | 1 | 128.10.e.a | 34 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.10.e.a | ✓ | 34 | 1.a | even | 1 | 1 | trivial |
16.10.e.a | ✓ | 34 | 16.e | even | 4 | 1 | inner |
64.10.e.a | 34 | 4.b | odd | 2 | 1 | ||
64.10.e.a | 34 | 16.f | odd | 4 | 1 | ||
128.10.e.a | 34 | 8.d | odd | 2 | 1 | ||
128.10.e.a | 34 | 16.f | odd | 4 | 1 | ||
128.10.e.b | 34 | 8.b | even | 2 | 1 | ||
128.10.e.b | 34 | 16.e | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(16, [\chi])\).