Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.28");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
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: | \(14.9360392488\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
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 | − | 43.1405i | − | 202.470i | −1349.11 | 1768.78 | −8734.66 | −2939.41 | 36113.2i | −21311.1 | − | 76306.1i | |||||||||||||||
28.2 | − | 39.9552i | 171.812i | −1084.42 | 917.921 | 6864.79 | 835.627 | 22871.2i | −9836.35 | − | 36675.8i | ||||||||||||||||
28.3 | − | 37.8978i | − | 104.304i | −924.245 | −2682.94 | −3952.89 | 4793.67 | 15623.2i | 8803.72 | 101678.i | ||||||||||||||||
28.4 | − | 34.5490i | 59.9553i | −681.634 | −636.317 | 2071.40 | −7630.94 | 5860.68i | 16088.4 | 21984.1i | |||||||||||||||||
28.5 | − | 27.8229i | − | 61.9183i | −262.115 | 1301.46 | −1722.75 | 8294.02 | − | 6952.53i | 15849.1 | − | 36210.3i | ||||||||||||||
28.6 | − | 25.3186i | 226.177i | −129.033 | −952.179 | 5726.51 | 6455.42 | − | 9696.20i | −31473.3 | 24107.9i | ||||||||||||||||
28.7 | − | 20.3455i | − | 80.4219i | 98.0611 | 535.165 | −1636.22 | −10636.4 | − | 12412.0i | 13215.3 | − | 10888.2i | ||||||||||||||
28.8 | − | 19.5556i | − | 267.976i | 129.577 | −595.738 | −5240.44 | −893.851 | − | 12546.4i | −52128.1 | 11650.0i | |||||||||||||||
28.9 | − | 10.3669i | 137.845i | 404.527 | −2029.73 | 1429.03 | −4871.43 | − | 9501.57i | 681.666 | 21042.0i | ||||||||||||||||
28.10 | − | 9.11441i | 171.314i | 428.927 | 2218.10 | 1561.43 | −1871.46 | − | 8576.00i | −9665.49 | − | 20216.7i | |||||||||||||||
28.11 | − | 6.67374i | − | 77.6458i | 467.461 | −531.517 | −518.188 | 5986.73 | − | 6536.67i | 13654.1 | 3547.20i | |||||||||||||||
28.12 | 6.67374i | 77.6458i | 467.461 | −531.517 | −518.188 | 5986.73 | 6536.67i | 13654.1 | − | 3547.20i | |||||||||||||||||
28.13 | 9.11441i | − | 171.314i | 428.927 | 2218.10 | 1561.43 | −1871.46 | 8576.00i | −9665.49 | 20216.7i | |||||||||||||||||
28.14 | 10.3669i | − | 137.845i | 404.527 | −2029.73 | 1429.03 | −4871.43 | 9501.57i | 681.666 | − | 21042.0i | ||||||||||||||||
28.15 | 19.5556i | 267.976i | 129.577 | −595.738 | −5240.44 | −893.851 | 12546.4i | −52128.1 | − | 11650.0i | |||||||||||||||||
28.16 | 20.3455i | 80.4219i | 98.0611 | 535.165 | −1636.22 | −10636.4 | 12412.0i | 13215.3 | 10888.2i | ||||||||||||||||||
28.17 | 25.3186i | − | 226.177i | −129.033 | −952.179 | 5726.51 | 6455.42 | 9696.20i | −31473.3 | − | 24107.9i | ||||||||||||||||
28.18 | 27.8229i | 61.9183i | −262.115 | 1301.46 | −1722.75 | 8294.02 | 6952.53i | 15849.1 | 36210.3i | ||||||||||||||||||
28.19 | 34.5490i | − | 59.9553i | −681.634 | −636.317 | 2071.40 | −7630.94 | − | 5860.68i | 16088.4 | − | 21984.1i | |||||||||||||||
28.20 | 37.8978i | 104.304i | −924.245 | −2682.94 | −3952.89 | 4793.67 | − | 15623.2i | 8803.72 | − | 101678.i | ||||||||||||||||
See all 22 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 | 29.10.b.a | ✓ | 22 |
29.b | even | 2 | 1 | inner | 29.10.b.a | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.10.b.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
29.10.b.a | ✓ | 22 | 29.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(29, [\chi])\).