Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,16,Mod(28,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.28");
S:= CuspForms(chi, 16);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.b (of order \(2\), degree \(1\), 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: | \(41.3811164790\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 | − | 350.198i | 3847.09i | −89870.4 | 106507. | 1.34724e6 | −2.76298e6 | 1.99971e7i | −451190. | − | 3.72984e7i | ||||||||||||||||
28.2 | − | 327.159i | − | 2163.79i | −74264.8 | 235169. | −707903. | 2.51260e6 | 1.35760e7i | 9.66691e6 | − | 7.69374e7i | |||||||||||||||
28.3 | − | 317.689i | − | 3423.11i | −68158.0 | −176108. | −1.08748e6 | 98456.3 | 1.12430e7i | 2.63125e6 | 5.59476e7i | ||||||||||||||||
28.4 | − | 297.362i | 5034.99i | −55656.3 | −122226. | 1.49722e6 | 2.97082e6 | 6.80612e6i | −1.10022e7 | 3.63455e7i | |||||||||||||||||
28.5 | − | 285.356i | − | 6609.41i | −48660.2 | 173971. | −1.88604e6 | −3.97639e6 | 4.53494e6i | −2.93355e7 | − | 4.96437e7i | |||||||||||||||
28.6 | − | 255.661i | 4338.48i | −32594.8 | −284737. | 1.10918e6 | −3.50246e6 | − | 44288.6i | −4.47354e6 | 7.27962e7i | ||||||||||||||||
28.7 | − | 240.042i | 2267.47i | −24852.3 | 195163. | 544290. | 168543. | − | 1.90009e6i | 9.20747e6 | − | 4.68475e7i | |||||||||||||||
28.8 | − | 231.586i | − | 885.668i | −20863.9 | −97507.9 | −205108. | 415555. | − | 2.75682e6i | 1.35645e7 | 2.25814e7i | |||||||||||||||
28.9 | − | 202.124i | − | 5797.90i | −8086.11 | 172967. | −1.17189e6 | 3.88134e6 | − | 4.98880e6i | −1.92667e7 | − | 3.49608e7i | ||||||||||||||
28.10 | − | 195.247i | 7180.80i | −5353.24 | 274602. | 1.40203e6 | −964261. | − | 5.35264e6i | −3.72150e7 | − | 5.36151e7i | |||||||||||||||
28.11 | − | 169.905i | − | 817.971i | 3900.43 | 58678.2 | −138977. | −2.72949e6 | − | 6.23013e6i | 1.36798e7 | − | 9.96969e6i | ||||||||||||||
28.12 | − | 150.477i | − | 6169.52i | 10124.7 | −189529. | −928371. | 634911. | − | 6.45436e6i | −2.37141e7 | 2.85197e7i | |||||||||||||||
28.13 | − | 118.325i | 6445.47i | 18767.1 | −115158. | 762661. | 493995. | − | 6.09791e6i | −2.71951e7 | 1.36261e7i | ||||||||||||||||
28.14 | − | 101.334i | 71.6384i | 22499.4 | −295367. | 7259.42 | 1.06773e6 | − | 5.60048e6i | 1.43438e7 | 2.99307e7i | ||||||||||||||||
28.15 | − | 84.1317i | 2443.60i | 25689.9 | 22406.3 | 205584. | 3.47198e6 | − | 4.91816e6i | 8.37775e6 | − | 1.88508e6i | |||||||||||||||
28.16 | − | 68.3669i | − | 2973.41i | 28094.0 | 302546. | −203283. | −153099. | − | 4.16094e6i | 5.50774e6 | − | 2.06841e7i | ||||||||||||||
28.17 | − | 14.2479i | 3944.08i | 32565.0 | −125301. | 56194.9 | −3.40529e6 | − | 930860.i | −1.20682e6 | 1.78527e6i | ||||||||||||||||
28.18 | − | 10.1196i | − | 4604.16i | 32665.6 | 98965.0 | −46592.0 | −632277. | − | 662159.i | −6.84936e6 | − | 1.00148e6i | ||||||||||||||
28.19 | 10.1196i | 4604.16i | 32665.6 | 98965.0 | −46592.0 | −632277. | 662159.i | −6.84936e6 | 1.00148e6i | ||||||||||||||||||
28.20 | 14.2479i | − | 3944.08i | 32565.0 | −125301. | 56194.9 | −3.40529e6 | 930860.i | −1.20682e6 | − | 1.78527e6i | ||||||||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.16.b.a | ✓ | 36 |
29.b | even | 2 | 1 | inner | 29.16.b.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.16.b.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
29.16.b.a | ✓ | 36 | 29.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(29, [\chi])\).