Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [668,2,Mod(9,668)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(668, base_ring=CyclotomicField(166))
chi = DirichletCharacter(H, H._module([0, 22]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("668.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 668 = 2^{2} \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 668.e (of order \(83\), degree \(82\), 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: | \(5.33400685502\) |
Analytic rank: | \(0\) |
Dimension: | \(1148\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{83})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{83}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | 0 | −0.426508 | − | 3.20063i | 0 | −3.18243 | − | 1.40730i | 0 | 1.55798 | − | 3.35028i | 0 | −7.16683 | + | 1.94460i | 0 | ||||||||||
9.2 | 0 | −0.360076 | − | 2.70211i | 0 | 2.66599 | + | 1.17892i | 0 | 0.359959 | − | 0.774059i | 0 | −4.27644 | + | 1.16034i | 0 | ||||||||||
9.3 | 0 | −0.342745 | − | 2.57205i | 0 | −0.628586 | − | 0.277966i | 0 | −1.47454 | + | 3.17087i | 0 | −3.60268 | + | 0.977525i | 0 | ||||||||||
9.4 | 0 | −0.198520 | − | 1.48975i | 0 | −0.0282706 | − | 0.0125015i | 0 | 2.06702 | − | 4.44493i | 0 | 0.715369 | − | 0.194103i | 0 | ||||||||||
9.5 | 0 | −0.178407 | − | 1.33882i | 0 | −1.62557 | − | 0.718841i | 0 | −0.315447 | + | 0.678340i | 0 | 1.13471 | − | 0.307883i | 0 | ||||||||||
9.6 | 0 | −0.0387535 | − | 0.290817i | 0 | −1.08122 | − | 0.478126i | 0 | −0.661531 | + | 1.42256i | 0 | 2.81224 | − | 0.763053i | 0 | ||||||||||
9.7 | 0 | −0.0142693 | − | 0.107081i | 0 | 2.37141 | + | 1.04866i | 0 | −0.709794 | + | 1.52635i | 0 | 2.88405 | − | 0.782538i | 0 | ||||||||||
9.8 | 0 | 0.0433273 | + | 0.325140i | 0 | 3.41908 | + | 1.51194i | 0 | −1.67119 | + | 3.59374i | 0 | 2.79147 | − | 0.757419i | 0 | ||||||||||
9.9 | 0 | 0.174236 | + | 1.30752i | 0 | −3.98478 | − | 1.76210i | 0 | −0.546271 | + | 1.17470i | 0 | 1.21607 | − | 0.329959i | 0 | ||||||||||
9.10 | 0 | 0.189949 | + | 1.42543i | 0 | 2.53109 | + | 1.11927i | 0 | 0.927279 | − | 1.99403i | 0 | 0.899535 | − | 0.244073i | 0 | ||||||||||
9.11 | 0 | 0.200238 | + | 1.50264i | 0 | −0.247141 | − | 0.109288i | 0 | 1.16135 | − | 2.49738i | 0 | 0.677472 | − | 0.183820i | 0 | ||||||||||
9.12 | 0 | 0.214009 | + | 1.60599i | 0 | −2.48108 | − | 1.09715i | 0 | 1.06258 | − | 2.28498i | 0 | 0.361919 | − | 0.0982005i | 0 | ||||||||||
9.13 | 0 | 0.353640 | + | 2.65381i | 0 | −1.27217 | − | 0.562563i | 0 | −1.92428 | + | 4.13797i | 0 | −4.02235 | + | 1.09139i | 0 | ||||||||||
9.14 | 0 | 0.383878 | + | 2.88073i | 0 | 1.71454 | + | 0.758185i | 0 | 0.166885 | − | 0.358871i | 0 | −5.25594 | + | 1.42611i | 0 | ||||||||||
21.1 | 0 | −3.30940 | − | 0.504914i | 0 | −2.26577 | − | 2.68874i | 0 | 1.34830 | − | 1.87063i | 0 | 7.83367 | + | 2.44733i | 0 | ||||||||||
21.2 | 0 | −2.93125 | − | 0.447219i | 0 | 1.46272 | + | 1.73577i | 0 | −0.690896 | + | 0.958545i | 0 | 5.52869 | + | 1.72722i | 0 | ||||||||||
21.3 | 0 | −1.90829 | − | 0.291147i | 0 | 1.01478 | + | 1.20422i | 0 | 2.44900 | − | 3.39773i | 0 | 0.693279 | + | 0.216588i | 0 | ||||||||||
21.4 | 0 | −1.77371 | − | 0.270614i | 0 | −1.21708 | − | 1.44428i | 0 | −0.212405 | + | 0.294689i | 0 | 0.209294 | + | 0.0653857i | 0 | ||||||||||
21.5 | 0 | −1.69992 | − | 0.259356i | 0 | −0.724427 | − | 0.859660i | 0 | −2.54038 | + | 3.52451i | 0 | −0.0410528 | − | 0.0128253i | 0 | ||||||||||
21.6 | 0 | −0.537235 | − | 0.0819657i | 0 | −1.19475 | − | 1.41779i | 0 | 0.821536 | − | 1.13980i | 0 | −2.58161 | − | 0.806524i | 0 | ||||||||||
See next 80 embeddings (of 1148 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
167.c | even | 83 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 668.2.e.a | ✓ | 1148 |
167.c | even | 83 | 1 | inner | 668.2.e.a | ✓ | 1148 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
668.2.e.a | ✓ | 1148 | 1.a | even | 1 | 1 | trivial |
668.2.e.a | ✓ | 1148 | 167.c | even | 83 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(668, [\chi])\).