Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [804,2,Mod(5,804)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 11, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("804.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 804 = 2^{2} \cdot 3 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 804.s (of order \(22\), degree \(10\), 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: | \(6.41997232251\) |
Analytic rank: | \(0\) |
Dimension: | \(200\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | 0 | −1.72660 | − | 0.137338i | 0 | 2.48625 | − | 0.730028i | 0 | 3.64531 | + | 3.15868i | 0 | 2.96228 | + | 0.474256i | 0 | ||||||||||
5.2 | 0 | −1.58909 | + | 0.689041i | 0 | 3.06595 | − | 0.900243i | 0 | 0.591666 | + | 0.512682i | 0 | 2.05045 | − | 2.18990i | 0 | ||||||||||
5.3 | 0 | −1.57919 | − | 0.711443i | 0 | −0.583807 | + | 0.171421i | 0 | −0.967947 | − | 0.838730i | 0 | 1.98770 | + | 2.24701i | 0 | ||||||||||
5.4 | 0 | −1.56138 | + | 0.749732i | 0 | −3.06595 | + | 0.900243i | 0 | 0.591666 | + | 0.512682i | 0 | 1.87580 | − | 2.34123i | 0 | ||||||||||
5.5 | 0 | −1.53236 | − | 0.807386i | 0 | −3.45254 | + | 1.01376i | 0 | −0.403229 | − | 0.349400i | 0 | 1.69626 | + | 2.47441i | 0 | ||||||||||
5.6 | 0 | −1.17979 | − | 1.26810i | 0 | 2.83063 | − | 0.831147i | 0 | −2.86249 | − | 2.48036i | 0 | −0.216176 | + | 2.99220i | 0 | ||||||||||
5.7 | 0 | −1.02689 | + | 1.39481i | 0 | −2.48625 | + | 0.730028i | 0 | 3.64531 | + | 3.15868i | 0 | −0.891005 | − | 2.86463i | 0 | ||||||||||
5.8 | 0 | −0.786808 | − | 1.54303i | 0 | 0.966140 | − | 0.283684i | 0 | 1.54930 | + | 1.34247i | 0 | −1.76187 | + | 2.42813i | 0 | ||||||||||
5.9 | 0 | −0.496478 | + | 1.65937i | 0 | 0.583807 | − | 0.171421i | 0 | −0.967947 | − | 0.838730i | 0 | −2.50702 | − | 1.64768i | 0 | ||||||||||
5.10 | 0 | −0.393301 | + | 1.68681i | 0 | 3.45254 | − | 1.01376i | 0 | −0.403229 | − | 0.349400i | 0 | −2.69063 | − | 1.32685i | 0 | ||||||||||
5.11 | 0 | 0.0434100 | − | 1.73151i | 0 | −1.09364 | + | 0.321121i | 0 | 2.09055 | + | 1.81147i | 0 | −2.99623 | − | 0.150329i | 0 | ||||||||||
5.12 | 0 | 0.185769 | + | 1.72206i | 0 | −2.83063 | + | 0.831147i | 0 | −2.86249 | − | 2.48036i | 0 | −2.93098 | + | 0.639810i | 0 | ||||||||||
5.13 | 0 | 0.429842 | − | 1.67787i | 0 | −3.03558 | + | 0.891328i | 0 | −1.25477 | − | 1.08727i | 0 | −2.63047 | − | 1.44244i | 0 | ||||||||||
5.14 | 0 | 0.575290 | − | 1.63372i | 0 | 2.05430 | − | 0.603197i | 0 | −2.91035 | − | 2.52183i | 0 | −2.33808 | − | 1.87973i | 0 | ||||||||||
5.15 | 0 | 0.650892 | + | 1.60510i | 0 | −0.966140 | + | 0.283684i | 0 | 1.54930 | + | 1.34247i | 0 | −2.15268 | + | 2.08949i | 0 | ||||||||||
5.16 | 0 | 1.33701 | + | 1.10109i | 0 | 1.09364 | − | 0.321121i | 0 | 2.09055 | + | 1.81147i | 0 | 0.575207 | + | 2.94434i | 0 | ||||||||||
5.17 | 0 | 1.48256 | − | 0.895546i | 0 | −2.22357 | + | 0.652900i | 0 | 2.01808 | + | 1.74867i | 0 | 1.39599 | − | 2.65541i | 0 | ||||||||||
5.18 | 0 | 1.54953 | + | 0.773916i | 0 | 3.03558 | − | 0.891328i | 0 | −1.25477 | − | 1.08727i | 0 | 1.80211 | + | 2.39842i | 0 | ||||||||||
5.19 | 0 | 1.61142 | + | 0.635084i | 0 | −2.05430 | + | 0.603197i | 0 | −2.91035 | − | 2.52183i | 0 | 2.19334 | + | 2.04677i | 0 | ||||||||||
5.20 | 0 | 1.64768 | − | 0.533989i | 0 | 2.22357 | − | 0.652900i | 0 | 2.01808 | + | 1.74867i | 0 | 2.42971 | − | 1.75969i | 0 | ||||||||||
See next 80 embeddings (of 200 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
67.f | odd | 22 | 1 | inner |
201.j | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 804.2.s.b | ✓ | 200 |
3.b | odd | 2 | 1 | inner | 804.2.s.b | ✓ | 200 |
67.f | odd | 22 | 1 | inner | 804.2.s.b | ✓ | 200 |
201.j | even | 22 | 1 | inner | 804.2.s.b | ✓ | 200 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
804.2.s.b | ✓ | 200 | 1.a | even | 1 | 1 | trivial |
804.2.s.b | ✓ | 200 | 3.b | odd | 2 | 1 | inner |
804.2.s.b | ✓ | 200 | 67.f | odd | 22 | 1 | inner |
804.2.s.b | ✓ | 200 | 201.j | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{200} + 60 T_{5}^{198} + 2469 T_{5}^{196} + 73443 T_{5}^{194} + 1818260 T_{5}^{192} + \cdots + 61\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).