Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,11,Mod(12,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.12");
S:= CuspForms(chi, 11);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.c (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: | \(18.4253603275\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −44.1507 | − | 44.1507i | −258.692 | − | 258.692i | 2874.56i | 3209.07i | 22842.8i | −14020.3 | 81703.6 | − | 81703.6i | 74794.0i | 141682. | − | 141682.i | ||||||||||
12.2 | −40.6077 | − | 40.6077i | 280.280 | + | 280.280i | 2273.97i | 1637.07i | − | 22763.0i | 25799.0 | 50758.2 | − | 50758.2i | 98064.8i | 66477.5 | − | 66477.5i | |||||||||
12.3 | −38.5476 | − | 38.5476i | −40.6574 | − | 40.6574i | 1947.84i | − | 4037.41i | 3134.49i | 12255.5 | 35611.7 | − | 35611.7i | − | 55743.0i | −155633. | + | 155633.i | ||||||||
12.4 | −36.5340 | − | 36.5340i | 128.848 | + | 128.848i | 1645.47i | − | 1229.81i | − | 9414.71i | −26326.6 | 22704.9 | − | 22704.9i | − | 25845.1i | −44929.8 | + | 44929.8i | |||||||
12.5 | −30.2185 | − | 30.2185i | −46.0531 | − | 46.0531i | 802.313i | 4207.29i | 2783.31i | 7518.31 | −6699.05 | + | 6699.05i | − | 54807.2i | 127138. | − | 127138.i | |||||||||
12.6 | −27.5873 | − | 27.5873i | −267.289 | − | 267.289i | 498.124i | − | 2802.28i | 14747.6i | 10208.5 | −14507.5 | + | 14507.5i | 83837.6i | −77307.4 | + | 77307.4i | |||||||||
12.7 | −17.7955 | − | 17.7955i | 303.416 | + | 303.416i | − | 390.637i | 2124.43i | − | 10798.9i | −21311.3 | −25174.2 | + | 25174.2i | 125074.i | 37805.4 | − | 37805.4i | ||||||||
12.8 | −16.3961 | − | 16.3961i | 189.226 | + | 189.226i | − | 486.333i | − | 5498.04i | − | 6205.14i | 9559.38 | −24763.6 | + | 24763.6i | 12563.6i | −90146.7 | + | 90146.7i | |||||||
12.9 | −16.1017 | − | 16.1017i | 130.144 | + | 130.144i | − | 505.467i | 1792.45i | − | 4191.09i | 14141.6 | −24627.1 | + | 24627.1i | − | 25174.1i | 28861.6 | − | 28861.6i | |||||||
12.10 | −15.8431 | − | 15.8431i | −149.513 | − | 149.513i | − | 521.995i | − | 249.743i | 4737.47i | −17394.4 | −24493.3 | + | 24493.3i | − | 14341.0i | −3956.70 | + | 3956.70i | |||||||
12.11 | −2.46716 | − | 2.46716i | −319.026 | − | 319.026i | − | 1011.83i | 5415.44i | 1574.18i | −6202.52 | −5022.70 | + | 5022.70i | 144507.i | 13360.8 | − | 13360.8i | |||||||||
12.12 | 1.96929 | + | 1.96929i | −66.5553 | − | 66.5553i | − | 1016.24i | − | 3811.80i | − | 262.134i | −21566.3 | 4017.83 | − | 4017.83i | − | 50189.8i | 7506.54 | − | 7506.54i | ||||||
12.13 | 4.04305 | + | 4.04305i | −142.774 | − | 142.774i | − | 991.307i | − | 351.494i | − | 1154.48i | 27728.5 | 8147.99 | − | 8147.99i | − | 18280.4i | 1421.11 | − | 1421.11i | ||||||
12.14 | 4.76401 | + | 4.76401i | 158.342 | + | 158.342i | − | 978.608i | 1411.74i | 1508.68i | 8006.05 | 9540.45 | − | 9540.45i | − | 8904.77i | −6725.56 | + | 6725.56i | ||||||||
12.15 | 8.23743 | + | 8.23743i | 85.7567 | + | 85.7567i | − | 888.289i | 5766.16i | 1412.83i | −20112.5 | 15752.4 | − | 15752.4i | − | 44340.6i | −47498.4 | + | 47498.4i | ||||||||
12.16 | 18.1033 | + | 18.1033i | 340.781 | + | 340.781i | − | 368.538i | − | 1274.81i | 12338.5i | 4091.64 | 25209.6 | − | 25209.6i | 173214.i | 23078.4 | − | 23078.4i | ||||||||
12.17 | 19.7480 | + | 19.7480i | 136.340 | + | 136.340i | − | 244.033i | − | 2681.56i | 5384.88i | −4320.20 | 25041.1 | − | 25041.1i | − | 21871.8i | 52955.4 | − | 52955.4i | |||||||
12.18 | 21.5476 | + | 21.5476i | −331.286 | − | 331.286i | − | 95.4007i | − | 4922.29i | − | 14276.9i | −7134.44 | 24120.4 | − | 24120.4i | 160452.i | 106064. | − | 106064.i | |||||||
12.19 | 27.4558 | + | 27.4558i | −172.516 | − | 172.516i | 483.643i | 2503.07i | − | 9473.14i | 5326.73 | 14835.9 | − | 14835.9i | 474.609i | −68723.9 | + | 68723.9i | |||||||||
12.20 | 29.1517 | + | 29.1517i | −96.9833 | − | 96.9833i | 675.640i | 1089.69i | − | 5654.45i | −17667.0 | 10155.3 | − | 10155.3i | − | 40237.5i | −31766.3 | + | 31766.3i | ||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.11.c.a | ✓ | 48 |
29.c | odd | 4 | 1 | inner | 29.11.c.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.11.c.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
29.11.c.a | ✓ | 48 | 29.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(29, [\chi])\).