Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [555,2,Mod(82,555)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("555.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 555 = 3 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 555.bd (of order \(12\), degree \(4\), 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: | \(4.43169731218\) |
Analytic rank: | \(0\) |
Dimension: | \(152\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −1.29013 | − | 2.23457i | −0.965926 | + | 0.258819i | −2.32886 | + | 4.03370i | 0.661101 | − | 2.13611i | 1.82452 | + | 1.82452i | −2.46133 | + | 0.659511i | 6.85757 | 0.866025 | − | 0.500000i | −5.62617 | + | 1.27857i | ||
82.2 | −1.28383 | − | 2.22366i | 0.965926 | − | 0.258819i | −2.29643 | + | 3.97754i | 2.20849 | − | 0.350110i | −1.81561 | − | 1.81561i | −4.23915 | + | 1.13588i | 6.65758 | 0.866025 | − | 0.500000i | −3.61385 | − | 4.46144i | ||
82.3 | −1.27502 | − | 2.20839i | −0.965926 | + | 0.258819i | −2.25133 | + | 3.89941i | −2.22698 | − | 0.201353i | 1.80314 | + | 1.80314i | 3.42337 | − | 0.917290i | 6.38185 | 0.866025 | − | 0.500000i | 2.39477 | + | 5.17478i | ||
82.4 | −1.23268 | − | 2.13507i | 0.965926 | − | 0.258819i | −2.03900 | + | 3.53166i | −2.23494 | − | 0.0709518i | −1.74327 | − | 1.74327i | −0.0512180 | + | 0.0137238i | 5.12305 | 0.866025 | − | 0.500000i | 2.60348 | + | 4.85921i | ||
82.5 | −1.19267 | − | 2.06576i | −0.965926 | + | 0.258819i | −1.84490 | + | 3.19547i | 0.0771478 | + | 2.23474i | 1.68668 | + | 1.68668i | −0.615378 | + | 0.164890i | 4.03075 | 0.866025 | − | 0.500000i | 4.52441 | − | 2.82466i | ||
82.6 | −1.15271 | − | 1.99655i | 0.965926 | − | 0.258819i | −1.65747 | + | 2.87082i | 1.97430 | + | 1.04983i | −1.63017 | − | 1.63017i | 2.40277 | − | 0.643820i | 3.03146 | 0.866025 | − | 0.500000i | −0.179756 | − | 5.15193i | ||
82.7 | −0.936247 | − | 1.62163i | 0.965926 | − | 0.258819i | −0.753118 | + | 1.30444i | −1.36154 | + | 1.77375i | −1.32405 | − | 1.32405i | 1.92977 | − | 0.517080i | −0.924569 | 0.866025 | − | 0.500000i | 4.15111 | + | 0.547248i | ||
82.8 | −0.774878 | − | 1.34213i | −0.965926 | + | 0.258819i | −0.200873 | + | 0.347922i | 0.286465 | − | 2.21764i | 1.09584 | + | 1.09584i | 0.817142 | − | 0.218952i | −2.47691 | 0.866025 | − | 0.500000i | −3.19834 | + | 1.33393i | ||
82.9 | −0.771950 | − | 1.33706i | −0.965926 | + | 0.258819i | −0.191813 | + | 0.332230i | 0.201731 | + | 2.22695i | 1.09170 | + | 1.09170i | 4.70754 | − | 1.26138i | −2.49552 | 0.866025 | − | 0.500000i | 2.82183 | − | 1.98882i | ||
82.10 | −0.723485 | − | 1.25311i | 0.965926 | − | 0.258819i | −0.0468616 | + | 0.0811666i | 1.66683 | − | 1.49053i | −1.02316 | − | 1.02316i | 1.20508 | − | 0.322900i | −2.75833 | 0.866025 | − | 0.500000i | −3.07373 | − | 1.01035i | ||
82.11 | −0.711025 | − | 1.23153i | 0.965926 | − | 0.258819i | −0.0111120 | + | 0.0192466i | 0.246144 | + | 2.22248i | −1.00554 | − | 1.00554i | −4.94627 | + | 1.32535i | −2.81249 | 0.866025 | − | 0.500000i | 2.56204 | − | 1.88337i | ||
82.12 | −0.702715 | − | 1.21714i | −0.965926 | + | 0.258819i | 0.0123832 | − | 0.0214483i | 2.21949 | + | 0.271757i | 0.993789 | + | 0.993789i | −3.44735 | + | 0.923714i | −2.84567 | 0.866025 | − | 0.500000i | −1.22890 | − | 2.89240i | ||
82.13 | −0.567625 | − | 0.983155i | −0.965926 | + | 0.258819i | 0.355604 | − | 0.615925i | −2.08976 | + | 0.795543i | 0.802743 | + | 0.802743i | −0.353460 | + | 0.0947093i | −3.07790 | 0.866025 | − | 0.500000i | 1.96834 | + | 1.60299i | ||
82.14 | −0.454649 | − | 0.787476i | 0.965926 | − | 0.258819i | 0.586588 | − | 1.01600i | −1.51872 | − | 1.64118i | −0.642971 | − | 0.642971i | 4.76595 | − | 1.27703i | −2.88537 | 0.866025 | − | 0.500000i | −0.601903 | + | 1.94212i | ||
82.15 | −0.343761 | − | 0.595411i | 0.965926 | − | 0.258819i | 0.763657 | − | 1.32269i | 2.00349 | + | 0.992985i | −0.486151 | − | 0.486151i | 0.629862 | − | 0.168771i | −2.42510 | 0.866025 | − | 0.500000i | −0.0974877 | − | 1.53425i | ||
82.16 | −0.261270 | − | 0.452533i | 0.965926 | − | 0.258819i | 0.863476 | − | 1.49558i | −2.05205 | − | 0.888300i | −0.369492 | − | 0.369492i | −3.40688 | + | 0.912870i | −1.94748 | 0.866025 | − | 0.500000i | 0.134155 | + | 1.16071i | ||
82.17 | −0.224905 | − | 0.389547i | −0.965926 | + | 0.258819i | 0.898836 | − | 1.55683i | −0.245397 | + | 2.22256i | 0.318064 | + | 0.318064i | −2.19780 | + | 0.588898i | −1.70823 | 0.866025 | − | 0.500000i | 0.920982 | − | 0.404272i | ||
82.18 | −0.0903978 | − | 0.156574i | −0.965926 | + | 0.258819i | 0.983656 | − | 1.70374i | −1.59092 | − | 1.57130i | 0.127842 | + | 0.127842i | −2.75537 | + | 0.738298i | −0.717272 | 0.866025 | − | 0.500000i | −0.102209 | + | 0.391137i | ||
82.19 | 0.0276845 | + | 0.0479509i | −0.965926 | + | 0.258819i | 0.998467 | − | 1.72940i | 2.23456 | − | 0.0821691i | −0.0391518 | − | 0.0391518i | 2.39340 | − | 0.641311i | 0.221306 | 0.866025 | − | 0.500000i | 0.0658027 | + | 0.104874i | ||
82.20 | 0.0430209 | + | 0.0745144i | −0.965926 | + | 0.258819i | 0.996298 | − | 1.72564i | 0.208099 | − | 2.22636i | −0.0608408 | − | 0.0608408i | 2.96943 | − | 0.795658i | 0.343530 | 0.866025 | − | 0.500000i | 0.174849 | − | 0.0802738i | ||
See next 80 embeddings (of 152 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.p | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 555.2.bd.a | ✓ | 152 |
5.c | odd | 4 | 1 | 555.2.bn.a | yes | 152 | |
37.g | odd | 12 | 1 | 555.2.bn.a | yes | 152 | |
185.p | even | 12 | 1 | inner | 555.2.bd.a | ✓ | 152 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
555.2.bd.a | ✓ | 152 | 1.a | even | 1 | 1 | trivial |
555.2.bd.a | ✓ | 152 | 185.p | even | 12 | 1 | inner |
555.2.bn.a | yes | 152 | 5.c | odd | 4 | 1 | |
555.2.bn.a | yes | 152 | 37.g | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).