Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [180,3,Mod(23,180)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(180, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 10, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("180.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 180.v (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: | \(4.90464475849\) |
Analytic rank: | \(0\) |
Dimension: | \(272\) |
Relative dimension: | \(68\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.99882 | + | 0.0686169i | −0.866619 | − | 2.87210i | 3.99058 | − | 0.274306i | −1.69193 | − | 4.70504i | 1.92929 | + | 5.68136i | 2.60652 | − | 9.72768i | −7.95765 | + | 0.822110i | −7.49794 | + | 4.97804i | 3.70472 | + | 9.28844i |
23.2 | −1.99843 | + | 0.0792022i | 2.33394 | + | 1.88487i | 3.98745 | − | 0.316560i | 3.77346 | + | 3.28040i | −4.81350 | − | 3.58193i | −3.10586 | + | 11.5912i | −7.94358 | + | 0.948438i | 1.89452 | + | 8.79834i | −7.80081 | − | 6.25679i |
23.3 | −1.98234 | − | 0.265231i | 1.57971 | − | 2.55040i | 3.85930 | + | 1.05155i | −4.16508 | + | 2.76625i | −3.80795 | + | 4.63676i | −0.704888 | + | 2.63068i | −7.37153 | − | 3.10814i | −4.00906 | − | 8.05776i | 8.99027 | − | 4.37893i |
23.4 | −1.97810 | − | 0.295138i | −2.70010 | + | 1.30745i | 3.82579 | + | 1.16763i | −2.81240 | − | 4.13406i | 5.72696 | − | 1.78938i | −1.42055 | + | 5.30156i | −7.22319 | − | 3.43882i | 5.58113 | − | 7.06053i | 4.34310 | + | 9.00764i |
23.5 | −1.96254 | + | 0.385279i | −2.90870 | + | 0.734496i | 3.70312 | − | 1.51225i | 0.178560 | + | 4.99681i | 5.42544 | − | 2.56214i | 0.293647 | − | 1.09590i | −6.68488 | + | 4.39458i | 7.92103 | − | 4.27285i | −2.27560 | − | 9.73764i |
23.6 | −1.94032 | − | 0.484928i | −0.800208 | + | 2.89131i | 3.52969 | + | 1.88183i | 4.85522 | − | 1.19451i | 2.95474 | − | 5.22202i | 2.17593 | − | 8.12068i | −5.93618 | − | 5.36301i | −7.71933 | − | 4.62730i | −9.99993 | − | 0.0367095i |
23.7 | −1.88318 | + | 0.673514i | 2.99492 | + | 0.174594i | 3.09276 | − | 2.53670i | −4.59989 | − | 1.95984i | −5.75757 | + | 1.68832i | −1.17779 | + | 4.39555i | −4.11573 | + | 6.86008i | 8.93903 | + | 1.04579i | 9.98242 | + | 0.592652i |
23.8 | −1.88005 | − | 0.682202i | −1.02400 | − | 2.81983i | 3.06920 | + | 2.56515i | 4.94759 | + | 0.722068i | 0.00147966 | + | 6.00000i | −2.05019 | + | 7.65143i | −4.02031 | − | 6.91644i | −6.90286 | + | 5.77500i | −8.80913 | − | 4.73278i |
23.9 | −1.84909 | + | 0.762148i | 0.204521 | + | 2.99302i | 2.83826 | − | 2.81856i | −4.76159 | + | 1.52554i | −2.65930 | − | 5.37849i | 2.15711 | − | 8.05045i | −3.10004 | + | 7.37494i | −8.91634 | + | 1.22427i | 7.64192 | − | 6.44989i |
23.10 | −1.83285 | + | 0.800411i | 2.64894 | − | 1.40823i | 2.71869 | − | 2.93407i | 4.99999 | + | 0.00771995i | −3.72794 | + | 4.70132i | 2.27318 | − | 8.48361i | −2.63449 | + | 7.55377i | 5.03375 | − | 7.46065i | −9.17042 | + | 3.98790i |
23.11 | −1.76475 | − | 0.941089i | 2.81520 | + | 1.03666i | 2.22870 | + | 3.32158i | −1.08683 | + | 4.88045i | −3.99253 | − | 4.47881i | 2.90708 | − | 10.8494i | −0.807202 | − | 7.95917i | 6.85065 | + | 5.83683i | 6.51093 | − | 7.58998i |
23.12 | −1.74814 | + | 0.971592i | −2.72685 | − | 1.25071i | 2.11202 | − | 3.39697i | 4.18642 | − | 2.73385i | 5.98211 | − | 0.462971i | −1.08898 | + | 4.06412i | −0.391643 | + | 7.99041i | 5.87146 | + | 6.82100i | −4.66227 | + | 8.84665i |
23.13 | −1.69861 | − | 1.05581i | 2.75442 | − | 1.18876i | 1.77052 | + | 3.58682i | 1.89315 | − | 4.62774i | −5.93379 | − | 0.888911i | −0.668907 | + | 2.49640i | 0.779590 | − | 7.96192i | 6.17368 | − | 6.54872i | −8.10174 | + | 5.86189i |
23.14 | −1.61680 | + | 1.17726i | −0.319431 | + | 2.98295i | 1.22810 | − | 3.80680i | 2.64586 | − | 4.24257i | −2.99525 | − | 5.19889i | −1.90052 | + | 7.09282i | 2.49601 | + | 7.60065i | −8.79593 | − | 1.90569i | 0.716781 | + | 9.97428i |
23.15 | −1.60157 | − | 1.19790i | −2.57707 | − | 1.53581i | 1.13006 | + | 3.83705i | −3.99036 | + | 3.01281i | 2.28762 | + | 5.54678i | −0.0614024 | + | 0.229157i | 2.78654 | − | 7.49901i | 4.28260 | + | 7.91576i | 9.99990 | − | 0.0451586i |
23.16 | −1.60019 | + | 1.19975i | −0.110713 | − | 2.99796i | 1.12121 | − | 3.83965i | 0.524705 | + | 4.97239i | 3.77396 | + | 4.66447i | −0.633651 | + | 2.36482i | 2.81247 | + | 7.48933i | −8.97549 | + | 0.663825i | −6.80525 | − | 7.32725i |
23.17 | −1.58088 | − | 1.22509i | 1.16828 | + | 2.76317i | 0.998333 | + | 3.87341i | −4.13729 | − | 2.80764i | 1.53822 | − | 5.79947i | −0.967290 | + | 3.60997i | 3.16702 | − | 7.34643i | −6.27025 | + | 6.45631i | 3.10094 | + | 9.50706i |
23.18 | −1.30527 | − | 1.51535i | −1.56620 | + | 2.55871i | −0.592542 | + | 3.95587i | 1.35415 | + | 4.81313i | 5.92165 | − | 0.966477i | −0.979164 | + | 3.65429i | 6.76793 | − | 4.26557i | −4.09403 | − | 8.01492i | 5.52602 | − | 8.33445i |
23.19 | −1.20757 | + | 1.59429i | −2.67197 | − | 1.36403i | −1.08354 | − | 3.85045i | −4.99807 | + | 0.138956i | 5.40126 | − | 2.61274i | 1.71830 | − | 6.41279i | 7.44719 | + | 2.92221i | 5.27885 | + | 7.28929i | 5.81399 | − | 8.13619i |
23.20 | −1.13981 | − | 1.64342i | −2.97923 | − | 0.352382i | −1.40165 | + | 3.74638i | 2.14586 | − | 4.51611i | 2.81666 | + | 5.29778i | 3.02532 | − | 11.2907i | 7.75450 | − | 1.96667i | 8.75165 | + | 2.09965i | −9.86775 | + | 1.62098i |
See next 80 embeddings (of 272 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
20.e | even | 4 | 1 | inner |
36.h | even | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
180.v | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 180.3.v.a | ✓ | 272 |
4.b | odd | 2 | 1 | inner | 180.3.v.a | ✓ | 272 |
5.c | odd | 4 | 1 | inner | 180.3.v.a | ✓ | 272 |
9.d | odd | 6 | 1 | inner | 180.3.v.a | ✓ | 272 |
20.e | even | 4 | 1 | inner | 180.3.v.a | ✓ | 272 |
36.h | even | 6 | 1 | inner | 180.3.v.a | ✓ | 272 |
45.l | even | 12 | 1 | inner | 180.3.v.a | ✓ | 272 |
180.v | odd | 12 | 1 | inner | 180.3.v.a | ✓ | 272 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
180.3.v.a | ✓ | 272 | 1.a | even | 1 | 1 | trivial |
180.3.v.a | ✓ | 272 | 4.b | odd | 2 | 1 | inner |
180.3.v.a | ✓ | 272 | 5.c | odd | 4 | 1 | inner |
180.3.v.a | ✓ | 272 | 9.d | odd | 6 | 1 | inner |
180.3.v.a | ✓ | 272 | 20.e | even | 4 | 1 | inner |
180.3.v.a | ✓ | 272 | 36.h | even | 6 | 1 | inner |
180.3.v.a | ✓ | 272 | 45.l | even | 12 | 1 | inner |
180.3.v.a | ✓ | 272 | 180.v | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(180, [\chi])\).