Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,11,Mod(58,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.58");
S:= CuspForms(chi, 11);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.c (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: | \(112.458233723\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
58.1 | − | 63.4704i | −140.296 | −3004.49 | −5840.70 | 8904.65i | 20957.7 | 125703.i | 19683.0 | 370712.i | |||||||||||||||||
58.2 | − | 63.4607i | −140.296 | −3003.25 | 3537.19 | 8903.28i | −15274.6 | 125605.i | 19683.0 | − | 224473.i | ||||||||||||||||
58.3 | − | 59.8859i | 140.296 | −2562.32 | 3689.76 | − | 8401.76i | 4188.49 | 92123.7i | 19683.0 | − | 220965.i | |||||||||||||||
58.4 | − | 59.5712i | 140.296 | −2524.73 | −4763.75 | − | 8357.61i | 8815.37 | 89400.5i | 19683.0 | 283782.i | ||||||||||||||||
58.5 | − | 58.9981i | −140.296 | −2456.77 | 878.334 | 8277.20i | −5578.77 | 84530.8i | 19683.0 | − | 51820.0i | ||||||||||||||||
58.6 | − | 58.2736i | 140.296 | −2371.81 | −453.456 | − | 8175.56i | 16053.8 | 78542.0i | 19683.0 | 26424.5i | ||||||||||||||||
58.7 | − | 56.3537i | −140.296 | −2151.73 | 3579.32 | 7906.20i | 7609.79 | 63551.9i | 19683.0 | − | 201708.i | ||||||||||||||||
58.8 | − | 55.7243i | 140.296 | −2081.20 | 5222.72 | − | 7817.90i | −1720.05 | 58911.7i | 19683.0 | − | 291033.i | |||||||||||||||
58.9 | − | 55.6543i | 140.296 | −2073.41 | −1651.80 | − | 7808.09i | −6427.48 | 58404.0i | 19683.0 | 91929.7i | ||||||||||||||||
58.10 | − | 54.8156i | −140.296 | −1980.75 | −4481.58 | 7690.41i | −27371.2 | 52444.5i | 19683.0 | 245660.i | |||||||||||||||||
58.11 | − | 53.2620i | −140.296 | −1812.84 | 3919.09 | 7472.46i | 33417.1 | 42015.4i | 19683.0 | − | 208739.i | ||||||||||||||||
58.12 | − | 51.4146i | −140.296 | −1619.46 | −2372.07 | 7213.27i | 15485.2 | 30615.5i | 19683.0 | 121959.i | |||||||||||||||||
58.13 | − | 50.8928i | 140.296 | −1566.07 | 2107.07 | − | 7140.06i | −26451.4 | 27587.7i | 19683.0 | − | 107235.i | |||||||||||||||
58.14 | − | 50.8045i | −140.296 | −1557.09 | 737.455 | 7127.67i | −21294.5 | 27083.5i | 19683.0 | − | 37466.0i | ||||||||||||||||
58.15 | − | 48.0671i | −140.296 | −1286.44 | −1212.73 | 6743.62i | −4240.63 | 12614.8i | 19683.0 | 58292.2i | |||||||||||||||||
58.16 | − | 47.5424i | 140.296 | −1236.28 | −4974.64 | − | 6670.02i | 12474.7 | 10092.4i | 19683.0 | 236507.i | ||||||||||||||||
58.17 | − | 46.8090i | 140.296 | −1167.08 | −1767.61 | − | 6567.12i | 23434.9 | 6697.50i | 19683.0 | 82740.0i | ||||||||||||||||
58.18 | − | 46.3046i | 140.296 | −1120.11 | 3858.39 | − | 6496.35i | 22459.5 | 4450.45i | 19683.0 | − | 178661.i | |||||||||||||||
58.19 | − | 45.0342i | 140.296 | −1004.08 | −5626.54 | − | 6318.12i | −22190.3 | − | 897.122i | 19683.0 | 253387.i | |||||||||||||||
58.20 | − | 44.5197i | 140.296 | −958.005 | −404.139 | − | 6245.94i | −20369.6 | − | 2938.10i | 19683.0 | 17992.1i | |||||||||||||||
See all 100 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.11.c.a | ✓ | 100 |
59.b | odd | 2 | 1 | inner | 177.11.c.a | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.11.c.a | ✓ | 100 | 1.a | even | 1 | 1 | trivial |
177.11.c.a | ✓ | 100 | 59.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(177, [\chi])\).