Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [133,2,Mod(10,133)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(133, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([3, 17]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("133.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 133 = 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 133.bf (of order \(18\), degree \(6\), 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: | \(1.06201034688\) |
Analytic rank: | \(0\) |
Dimension: | \(66\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −1.61995 | − | 1.93058i | 0.0453294 | + | 0.257076i | −0.755612 | + | 4.28529i | 4.11821 | − | 0.726151i | 0.422875 | − | 0.503963i | −0.270204 | + | 2.63192i | 5.13205 | − | 2.96299i | 2.75504 | − | 1.00275i | −8.07319 | − | 6.77421i |
10.2 | −1.52009 | − | 1.81157i | −0.439106 | − | 2.49029i | −0.623824 | + | 3.53788i | −1.04499 | + | 0.184260i | −3.84386 | + | 4.58094i | −0.591482 | − | 2.57879i | 3.26138 | − | 1.88296i | −3.18966 | + | 1.16094i | 1.92228 | + | 1.61298i |
10.3 | −1.27330 | − | 1.51746i | 0.476887 | + | 2.70456i | −0.334098 | + | 1.89477i | −2.20568 | + | 0.388921i | 3.49685 | − | 4.16738i | −2.50884 | + | 0.840077i | −0.130386 | + | 0.0752786i | −4.26815 | + | 1.55348i | 3.39868 | + | 2.85183i |
10.4 | −0.790508 | − | 0.942090i | 0.0284469 | + | 0.161330i | 0.0846645 | − | 0.480156i | 0.850790 | − | 0.150017i | 0.129500 | − | 0.154333i | −0.990946 | − | 2.45317i | −2.64937 | + | 1.52962i | 2.79386 | − | 1.01688i | −0.813886 | − | 0.682931i |
10.5 | −0.432162 | − | 0.515031i | −0.514014 | − | 2.91512i | 0.268804 | − | 1.52446i | 0.960858 | − | 0.169425i | −1.27924 | + | 1.52454i | −0.595102 | + | 2.57796i | −2.06581 | + | 1.19270i | −5.41462 | + | 1.97076i | −0.502506 | − | 0.421652i |
10.6 | −0.373593 | − | 0.445230i | 0.289267 | + | 1.64052i | 0.288638 | − | 1.63695i | 0.294690 | − | 0.0519618i | 0.622340 | − | 0.741676i | 2.63474 | + | 0.241108i | −1.84333 | + | 1.06425i | 0.211456 | − | 0.0769638i | −0.133229 | − | 0.111792i |
10.7 | 0.414484 | + | 0.493962i | −0.315845 | − | 1.79125i | 0.275094 | − | 1.56014i | −3.56788 | + | 0.629114i | 0.753896 | − | 0.898458i | 2.01676 | − | 1.71251i | 2.00154 | − | 1.15559i | −0.289728 | + | 0.105452i | −1.78959 | − | 1.50164i |
10.8 | 0.480891 | + | 0.573103i | −0.0906786 | − | 0.514264i | 0.250105 | − | 1.41842i | 1.59609 | − | 0.281434i | 0.251120 | − | 0.299273i | −2.47659 | + | 0.930864i | 2.22898 | − | 1.28690i | 2.56283 | − | 0.932795i | 0.928835 | + | 0.779385i |
10.9 | 0.818304 | + | 0.975217i | 0.347752 | + | 1.97220i | 0.0658701 | − | 0.373568i | −2.34435 | + | 0.413372i | −1.63876 | + | 1.95299i | 0.0571233 | + | 2.64513i | 2.62321 | − | 1.51451i | −0.949566 | + | 0.345614i | −2.32152 | − | 1.94798i |
10.10 | 1.48693 | + | 1.77205i | −0.429930 | − | 2.43826i | −0.581918 | + | 3.30022i | 0.765719 | − | 0.135017i | 3.68144 | − | 4.38737i | 1.90031 | + | 1.84088i | −2.70677 | + | 1.56275i | −2.94117 | + | 1.07050i | 1.37783 | + | 1.15613i |
10.11 | 1.54295 | + | 1.83882i | 0.0415830 | + | 0.235829i | −0.653260 | + | 3.70482i | −0.923449 | + | 0.162829i | −0.369487 | + | 0.440337i | −1.91360 | − | 1.82706i | −3.66282 | + | 2.11473i | 2.76519 | − | 1.00645i | −1.72425 | − | 1.44682i |
33.1 | −2.42618 | − | 0.427801i | 1.81066 | − | 0.659027i | 3.82394 | + | 1.39180i | −0.965340 | − | 2.65225i | −4.67492 | + | 0.824314i | 2.39627 | + | 1.12156i | −4.41506 | − | 2.54904i | 0.546042 | − | 0.458184i | 1.20745 | + | 6.84780i |
33.2 | −1.99215 | − | 0.351269i | −0.662476 | + | 0.241122i | 1.96587 | + | 0.715519i | 0.0536790 | + | 0.147482i | 1.40445 | − | 0.247642i | −1.17382 | − | 2.37111i | −0.161239 | − | 0.0930914i | −1.91740 | + | 1.60889i | −0.0551306 | − | 0.312661i |
33.3 | −1.84540 | − | 0.325394i | −2.57736 | + | 0.938083i | 1.42023 | + | 0.516922i | −0.854536 | − | 2.34782i | 5.06151 | − | 0.892480i | −1.55266 | + | 2.14225i | 0.792944 | + | 0.457807i | 3.46466 | − | 2.90719i | 0.812995 | + | 4.61072i |
33.4 | −0.928599 | − | 0.163737i | −0.187541 | + | 0.0682594i | −1.04390 | − | 0.379948i | 0.803754 | + | 2.20830i | 0.185327 | − | 0.0326782i | 2.64571 | − | 0.0147547i | 2.54034 | + | 1.46667i | −2.26762 | + | 1.90276i | −0.384786 | − | 2.18223i |
33.5 | −0.697238 | − | 0.122942i | 2.27698 | − | 0.828751i | −1.40836 | − | 0.512601i | −0.581955 | − | 1.59891i | −1.68948 | + | 0.297901i | −0.813635 | − | 2.51754i | 2.14522 | + | 1.23854i | 2.19966 | − | 1.84573i | 0.209188 | + | 1.18637i |
33.6 | −0.0226089 | − | 0.00398656i | −1.47792 | + | 0.537920i | −1.87889 | − | 0.683860i | 0.513048 | + | 1.40959i | 0.0355586 | − | 0.00626995i | −1.84251 | + | 1.89873i | 0.0795172 | + | 0.0459093i | −0.403236 | + | 0.338355i | −0.00598004 | − | 0.0339145i |
33.7 | 0.397723 | + | 0.0701294i | −1.42173 | + | 0.517468i | −1.72612 | − | 0.628256i | −1.40124 | − | 3.84988i | −0.601745 | + | 0.106104i | 2.51181 | − | 0.831138i | −1.34196 | − | 0.774783i | −0.544588 | + | 0.456963i | −0.287317 | − | 1.62946i |
33.8 | 1.19145 | + | 0.210084i | 1.79234 | − | 0.652360i | −0.503976 | − | 0.183432i | −0.404136 | − | 1.11035i | 2.27253 | − | 0.400709i | 0.366724 | + | 2.62021i | −2.65741 | − | 1.53425i | 0.488789 | − | 0.410143i | −0.248238 | − | 1.40783i |
33.9 | 1.20110 | + | 0.211786i | 1.44898 | − | 0.527386i | −0.481603 | − | 0.175289i | 1.20912 | + | 3.32203i | 1.85206 | − | 0.326568i | −1.34413 | − | 2.27889i | −2.65378 | − | 1.53216i | −0.476721 | + | 0.400016i | 0.748713 | + | 4.24616i |
See all 66 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
133.bf | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 133.2.bf.a | yes | 66 |
7.b | odd | 2 | 1 | 931.2.bj.a | 66 | ||
7.c | even | 3 | 1 | 931.2.be.a | 66 | ||
7.c | even | 3 | 1 | 931.2.bf.a | 66 | ||
7.d | odd | 6 | 1 | 133.2.bb.a | ✓ | 66 | |
7.d | odd | 6 | 1 | 931.2.be.b | 66 | ||
19.f | odd | 18 | 1 | 133.2.bb.a | ✓ | 66 | |
133.ba | even | 18 | 1 | 931.2.bf.a | 66 | ||
133.bb | even | 18 | 1 | 931.2.be.a | 66 | ||
133.bd | odd | 18 | 1 | 931.2.be.b | 66 | ||
133.be | odd | 18 | 1 | 931.2.bj.a | 66 | ||
133.bf | even | 18 | 1 | inner | 133.2.bf.a | yes | 66 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
133.2.bb.a | ✓ | 66 | 7.d | odd | 6 | 1 | |
133.2.bb.a | ✓ | 66 | 19.f | odd | 18 | 1 | |
133.2.bf.a | yes | 66 | 1.a | even | 1 | 1 | trivial |
133.2.bf.a | yes | 66 | 133.bf | even | 18 | 1 | inner |
931.2.be.a | 66 | 7.c | even | 3 | 1 | ||
931.2.be.a | 66 | 133.bb | even | 18 | 1 | ||
931.2.be.b | 66 | 7.d | odd | 6 | 1 | ||
931.2.be.b | 66 | 133.bd | odd | 18 | 1 | ||
931.2.bf.a | 66 | 7.c | even | 3 | 1 | ||
931.2.bf.a | 66 | 133.ba | even | 18 | 1 | ||
931.2.bj.a | 66 | 7.b | odd | 2 | 1 | ||
931.2.bj.a | 66 | 133.be | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(133, [\chi])\).