Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(92,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.92");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.cc (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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(36\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
92.1 | −0.713154 | − | 2.66153i | 1.69514 | + | 0.355654i | −4.84309 | + | 2.79616i | 1.19095 | + | 1.89252i | −0.262317 | − | 4.76530i | 0.965926 | − | 0.258819i | 6.99916 | + | 6.99916i | 2.74702 | + | 1.20577i | 4.18768 | − | 4.51940i |
92.2 | −0.679005 | − | 2.53408i | 0.297629 | + | 1.70629i | −4.22847 | + | 2.44131i | 0.0454078 | − | 2.23561i | 4.12178 | − | 1.91279i | −0.965926 | + | 0.258819i | 5.34747 | + | 5.34747i | −2.82283 | + | 1.01568i | −5.69604 | + | 1.40292i |
92.3 | −0.646610 | − | 2.41318i | −1.26505 | + | 1.18307i | −3.67328 | + | 2.12077i | −2.08010 | + | 0.820487i | 3.67295 | + | 2.28781i | 0.965926 | − | 0.258819i | 3.95984 | + | 3.95984i | 0.200708 | − | 2.99328i | 3.32499 | + | 4.48911i |
92.4 | −0.626308 | − | 2.33741i | −1.18606 | − | 1.26224i | −3.33919 | + | 1.92788i | −0.366266 | + | 2.20587i | −2.20755 | + | 3.56287i | −0.965926 | + | 0.258819i | 3.17541 | + | 3.17541i | −0.186522 | + | 2.99420i | 5.38542 | − | 0.525436i |
92.5 | −0.616561 | − | 2.30104i | 1.18411 | − | 1.26408i | −3.18258 | + | 1.83746i | −2.21436 | + | 0.310826i | −3.63877 | − | 1.94529i | −0.965926 | + | 0.258819i | 2.82137 | + | 2.82137i | −0.195791 | − | 2.99360i | 2.08051 | + | 4.90368i |
92.6 | −0.540409 | − | 2.01684i | 1.29595 | − | 1.14913i | −2.04353 | + | 1.17983i | −0.0970501 | − | 2.23396i | −3.01795 | − | 1.99273i | 0.965926 | − | 0.258819i | 0.531022 | + | 0.531022i | 0.358997 | − | 2.97844i | −4.45308 | + | 1.40299i |
92.7 | −0.522977 | − | 1.95178i | 0.00508642 | − | 1.73204i | −1.80388 | + | 1.04147i | 2.08954 | + | 0.796124i | −3.38322 | + | 0.895892i | 0.965926 | − | 0.258819i | 0.118505 | + | 0.118505i | −2.99995 | − | 0.0176198i | 0.461075 | − | 4.49468i |
92.8 | −0.484811 | − | 1.80934i | 0.413768 | + | 1.68190i | −1.30661 | + | 0.754373i | 1.61436 | + | 1.54720i | 2.84253 | − | 1.56405i | −0.965926 | + | 0.258819i | −0.650680 | − | 0.650680i | −2.65759 | + | 1.39183i | 2.01675 | − | 3.67103i |
92.9 | −0.372518 | − | 1.39026i | −1.46969 | + | 0.916517i | −0.0619937 | + | 0.0357921i | −0.217588 | + | 2.22546i | 1.82168 | + | 1.70183i | −0.965926 | + | 0.258819i | −1.96262 | − | 1.96262i | 1.31999 | − | 2.69400i | 3.17501 | − | 0.526519i |
92.10 | −0.352695 | − | 1.31628i | −0.672895 | − | 1.59600i | 0.123861 | − | 0.0715111i | −0.441766 | − | 2.19200i | −1.86345 | + | 1.44862i | −0.965926 | + | 0.258819i | −2.06498 | − | 2.06498i | −2.09442 | + | 2.14788i | −2.72946 | + | 1.35459i |
92.11 | −0.303557 | − | 1.13289i | −0.471331 | + | 1.66669i | 0.540756 | − | 0.312205i | −0.436585 | − | 2.19303i | 2.03125 | + | 0.0280322i | 0.965926 | − | 0.258819i | −2.17651 | − | 2.17651i | −2.55569 | − | 1.57112i | −2.35194 | + | 1.16031i |
92.12 | −0.292475 | − | 1.09153i | −1.63819 | − | 0.562423i | 0.626149 | − | 0.361507i | −2.16876 | − | 0.544513i | −0.134772 | + | 1.95264i | 0.965926 | − | 0.258819i | −2.17584 | − | 2.17584i | 2.36736 | + | 1.84272i | 0.0399539 | + | 2.52653i |
92.13 | −0.270870 | − | 1.01090i | 1.36595 | + | 1.06499i | 0.783502 | − | 0.452355i | 2.05901 | − | 0.872052i | 0.706600 | − | 1.66931i | 0.965926 | − | 0.258819i | −2.14957 | − | 2.14957i | 0.731614 | + | 2.90942i | −1.43928 | − | 1.84524i |
92.14 | −0.231820 | − | 0.865164i | 1.63235 | − | 0.579155i | 1.03728 | − | 0.598875i | −0.767760 | + | 2.10013i | −0.879477 | − | 1.27799i | 0.965926 | − | 0.258819i | −2.02528 | − | 2.02528i | 2.32916 | − | 1.89077i | 1.99494 | + | 0.177387i |
92.15 | −0.220022 | − | 0.821133i | 0.953832 | − | 1.44575i | 1.10620 | − | 0.638666i | 1.96491 | + | 1.06730i | −1.39702 | − | 0.465126i | −0.965926 | + | 0.258819i | −1.97004 | − | 1.97004i | −1.18041 | − | 2.75801i | 0.444073 | − | 1.84828i |
92.16 | −0.165403 | − | 0.617291i | −1.49090 | + | 0.881605i | 1.37836 | − | 0.795797i | 1.71710 | − | 1.43233i | 0.790805 | + | 0.774498i | −0.965926 | + | 0.258819i | −1.62300 | − | 1.62300i | 1.44555 | − | 2.62876i | −1.16818 | − | 0.823041i |
92.17 | −0.0549668 | − | 0.205139i | 1.73204 | − | 0.00719422i | 1.69299 | − | 0.977448i | −1.03445 | − | 1.98240i | −0.0966803 | − | 0.354913i | −0.965926 | + | 0.258819i | −0.593915 | − | 0.593915i | 2.99990 | − | 0.0249213i | −0.349807 | + | 0.321173i |
92.18 | −0.0242815 | − | 0.0906199i | −0.235889 | + | 1.71591i | 1.72443 | − | 0.995599i | −0.436480 | + | 2.19305i | 0.161224 | − | 0.0202887i | 0.965926 | − | 0.258819i | −0.264770 | − | 0.264770i | −2.88871 | − | 0.809530i | 0.209333 | − | 0.0136969i |
92.19 | 0.0573303 | + | 0.213960i | −1.05605 | − | 1.37287i | 1.68956 | − | 0.975467i | 1.86575 | − | 1.23247i | 0.233194 | − | 0.304659i | 0.965926 | − | 0.258819i | 0.618832 | + | 0.618832i | −0.769518 | + | 2.89963i | 0.370664 | + | 0.328536i |
92.20 | 0.0867243 | + | 0.323660i | −1.73132 | + | 0.0504811i | 1.63482 | − | 0.943862i | −2.20059 | + | 0.396767i | −0.166486 | − | 0.555979i | −0.965926 | + | 0.258819i | 0.921139 | + | 0.921139i | 2.99490 | − | 0.174797i | −0.319262 | − | 0.677831i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.cc.a | ✓ | 144 |
3.b | odd | 2 | 1 | 945.2.cf.a | 144 | ||
5.c | odd | 4 | 1 | inner | 315.2.cc.a | ✓ | 144 |
9.c | even | 3 | 1 | 945.2.cf.a | 144 | ||
9.d | odd | 6 | 1 | inner | 315.2.cc.a | ✓ | 144 |
15.e | even | 4 | 1 | 945.2.cf.a | 144 | ||
45.k | odd | 12 | 1 | 945.2.cf.a | 144 | ||
45.l | even | 12 | 1 | inner | 315.2.cc.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.cc.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
315.2.cc.a | ✓ | 144 | 5.c | odd | 4 | 1 | inner |
315.2.cc.a | ✓ | 144 | 9.d | odd | 6 | 1 | inner |
315.2.cc.a | ✓ | 144 | 45.l | even | 12 | 1 | inner |
945.2.cf.a | 144 | 3.b | odd | 2 | 1 | ||
945.2.cf.a | 144 | 9.c | even | 3 | 1 | ||
945.2.cf.a | 144 | 15.e | even | 4 | 1 | ||
945.2.cf.a | 144 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).