Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [109,8,Mod(108,109)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("109.108");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 109 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 109.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: | \(34.0499677778\) |
Analytic rank: | \(0\) |
Dimension: | \(62\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
108.1 | − | 22.1186i | 68.3096 | −361.232 | 198.755 | − | 1510.91i | 477.809 | 5158.77i | 2479.21 | − | 4396.19i | |||||||||||||||
108.2 | − | 21.7464i | −13.5748 | −344.905 | −10.6141 | 295.203i | −628.336 | 4716.91i | −2002.72 | 230.818i | |||||||||||||||||
108.3 | − | 20.6762i | −63.1733 | −299.506 | −321.718 | 1306.18i | 710.246 | 3546.09i | 1803.86 | 6651.91i | |||||||||||||||||
108.4 | − | 20.5650i | −83.1155 | −294.921 | 534.705 | 1709.27i | −279.478 | 3432.74i | 4721.18 | − | 10996.2i | ||||||||||||||||
108.5 | − | 19.1107i | 70.4891 | −237.218 | −288.848 | − | 1347.09i | −109.637 | 2087.22i | 2781.72 | 5520.08i | ||||||||||||||||
108.6 | − | 18.7221i | −3.68685 | −222.515 | 339.279 | 69.0254i | 552.193 | 1769.52i | −2173.41 | − | 6351.99i | ||||||||||||||||
108.7 | − | 18.2217i | −4.72274 | −204.029 | 113.191 | 86.0562i | 1550.44 | 1385.38i | −2164.70 | − | 2062.52i | ||||||||||||||||
108.8 | − | 18.1795i | 9.33049 | −202.493 | −443.908 | − | 169.623i | −988.125 | 1354.25i | −2099.94 | 8070.01i | ||||||||||||||||
108.9 | − | 17.9931i | 45.7905 | −195.751 | 340.028 | − | 823.912i | −1712.93 | 1219.06i | −90.2316 | − | 6118.15i | |||||||||||||||
108.10 | − | 17.5675i | 39.7962 | −180.616 | −277.314 | − | 699.119i | 1237.74 | 924.338i | −603.263 | 4871.71i | ||||||||||||||||
108.11 | − | 15.8282i | −77.0878 | −122.532 | −187.933 | 1220.16i | −1259.33 | − | 86.5528i | 3755.53 | 2974.64i | ||||||||||||||||
108.12 | − | 15.6070i | −36.0261 | −115.577 | 219.020 | 562.257i | −1106.83 | − | 193.886i | −889.123 | − | 3418.23i | |||||||||||||||
108.13 | − | 15.1160i | −59.8207 | −100.494 | 62.1458 | 904.251i | 706.973 | − | 415.786i | 1391.52 | − | 939.396i | |||||||||||||||
108.14 | − | 14.2491i | 65.6723 | −75.0363 | 367.355 | − | 935.771i | 501.747 | − | 754.684i | 2125.86 | − | 5234.48i | ||||||||||||||
108.15 | − | 12.0377i | 26.3124 | −16.9068 | 54.1389 | − | 316.741i | −225.072 | − | 1337.31i | −1494.66 | − | 651.710i | ||||||||||||||
108.16 | − | 12.0291i | −31.0591 | −16.6993 | −170.363 | 373.613i | −271.091 | − | 1338.85i | −1222.34 | 2049.32i | ||||||||||||||||
108.17 | − | 11.2548i | 88.3366 | 1.33012 | −147.950 | − | 994.209i | −1272.73 | − | 1455.58i | 5616.36 | 1665.14i | |||||||||||||||
108.18 | − | 10.9727i | −20.6182 | 7.60069 | −529.822 | 226.236i | 1447.54 | − | 1487.90i | −1761.89 | 5813.56i | ||||||||||||||||
108.19 | − | 9.83474i | 28.1069 | 31.2778 | −189.209 | − | 276.424i | −141.499 | − | 1566.46i | −1397.00 | 1860.82i | |||||||||||||||
108.20 | − | 8.61539i | −73.6630 | 53.7750 | 305.812 | 634.636i | 1462.18 | − | 1566.06i | 3239.24 | − | 2634.69i | |||||||||||||||
See all 62 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
109.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 109.8.b.a | ✓ | 62 |
109.b | even | 2 | 1 | inner | 109.8.b.a | ✓ | 62 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
109.8.b.a | ✓ | 62 | 1.a | even | 1 | 1 | trivial |
109.8.b.a | ✓ | 62 | 109.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(109, [\chi])\).