Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,3,Mod(10,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(58))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.10");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.g (of order \(58\), degree \(28\), 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: | \(4.82290067918\) |
Analytic rank: | \(0\) |
Dimension: | \(560\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{58})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{58}]$ |
$q$-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 | −3.55013 | − | 1.41450i | 1.72190 | + | 0.187268i | 7.69866 | + | 7.29256i | 2.93856 | − | 3.45954i | −5.84808 | − | 3.10045i | −1.56353 | − | 0.940746i | −10.5974 | − | 22.9059i | 2.92986 | + | 0.644911i | −15.3258 | + | 8.12523i |
10.2 | −3.48729 | − | 1.38946i | −1.72190 | − | 0.187268i | 7.32661 | + | 6.94014i | 0.951158 | − | 1.11979i | 5.74456 | + | 3.04557i | 1.10368 | + | 0.664063i | −9.60207 | − | 20.7545i | 2.92986 | + | 0.644911i | −4.87287 | + | 2.58343i |
10.3 | −2.66127 | − | 1.06035i | −1.72190 | − | 0.187268i | 3.05402 | + | 2.89292i | −5.63025 | + | 6.62844i | 4.38386 | + | 2.32417i | −9.66711 | − | 5.81651i | −0.248591 | − | 0.537320i | 2.92986 | + | 0.644911i | 22.0120 | − | 11.6700i |
10.4 | −2.31503 | − | 0.922394i | 1.72190 | + | 0.187268i | 1.60459 | + | 1.51995i | −0.698140 | + | 0.821913i | −3.81352 | − | 2.02180i | −4.78757 | − | 2.88058i | 1.87281 | + | 4.04801i | 2.92986 | + | 0.644911i | 2.37434 | − | 1.25880i |
10.5 | −2.17964 | − | 0.868447i | −1.72190 | − | 0.187268i | 1.09264 | + | 1.03500i | 5.68569 | − | 6.69371i | 3.59048 | + | 1.90355i | 0.838143 | + | 0.504294i | 2.45800 | + | 5.31287i | 2.92986 | + | 0.644911i | −18.2059 | + | 9.65215i |
10.6 | −2.06048 | − | 0.820970i | 1.72190 | + | 0.187268i | 0.667604 | + | 0.632388i | 4.71743 | − | 5.55378i | −3.39419 | − | 1.79949i | 8.57272 | + | 5.15803i | 2.86886 | + | 6.20094i | 2.92986 | + | 0.644911i | −14.2797 | + | 7.57059i |
10.7 | −1.49154 | − | 0.594282i | −1.72190 | − | 0.187268i | −1.03247 | − | 0.978012i | −2.12045 | + | 2.49638i | 2.45698 | + | 1.30261i | 6.56905 | + | 3.95247i | 3.65540 | + | 7.90102i | 2.92986 | + | 0.644911i | 4.64627 | − | 2.46330i |
10.8 | −1.45950 | − | 0.581520i | 1.72190 | + | 0.187268i | −1.11199 | − | 1.05334i | −2.61612 | + | 3.07994i | −2.40422 | − | 1.27464i | −1.72503 | − | 1.03791i | 3.64916 | + | 7.88752i | 2.92986 | + | 0.644911i | 5.60928 | − | 2.97385i |
10.9 | −0.911464 | − | 0.363160i | −1.72190 | − | 0.187268i | −2.20510 | − | 2.08878i | 0.969991 | − | 1.14196i | 1.50144 | + | 0.796012i | −4.53786 | − | 2.73034i | 2.89920 | + | 6.26652i | 2.92986 | + | 0.644911i | −1.29883 | + | 0.688594i |
10.10 | 0.0935508 | + | 0.0372741i | 1.72190 | + | 0.187268i | −2.89662 | − | 2.74382i | −5.55524 | + | 6.54013i | 0.154105 | + | 0.0817011i | 5.52684 | + | 3.32539i | −0.337844 | − | 0.730238i | 2.92986 | + | 0.644911i | −0.763474 | + | 0.404768i |
10.11 | 0.140866 | + | 0.0561262i | 1.72190 | + | 0.187268i | −2.88729 | − | 2.73499i | 1.82294 | − | 2.14613i | 0.232047 | + | 0.123023i | −8.41764 | − | 5.06473i | −0.507898 | − | 1.09780i | 2.92986 | + | 0.644911i | 0.377244 | − | 0.200002i |
10.12 | 0.782131 | + | 0.311630i | −1.72190 | − | 0.187268i | −2.38937 | − | 2.26333i | −2.21295 | + | 2.60528i | −1.28839 | − | 0.683062i | 9.58665 | + | 5.76810i | −2.57754 | − | 5.57127i | 2.92986 | + | 0.644911i | −2.54270 | + | 1.34805i |
10.13 | 0.819309 | + | 0.326442i | 1.72190 | + | 0.187268i | −2.33928 | − | 2.21588i | 1.32036 | − | 1.55445i | 1.34963 | + | 0.715530i | 7.54931 | + | 4.54227i | −2.67451 | − | 5.78087i | 2.92986 | + | 0.644911i | 1.58922 | − | 0.842551i |
10.14 | 1.61637 | + | 0.644021i | −1.72190 | − | 0.187268i | −0.706089 | − | 0.668843i | −2.42613 | + | 2.85626i | −2.66262 | − | 1.41163i | −4.88728 | − | 2.94058i | −3.63289 | − | 7.85237i | 2.92986 | + | 0.644911i | −5.76103 | + | 3.05430i |
10.15 | 1.89981 | + | 0.756953i | −1.72190 | − | 0.187268i | 0.132311 | + | 0.125332i | 3.62948 | − | 4.27295i | −3.12952 | − | 1.65917i | −2.21876 | − | 1.33498i | −3.27829 | − | 7.08590i | 2.92986 | + | 0.644911i | 10.1297 | − | 5.37045i |
10.16 | 2.13762 | + | 0.851704i | 1.72190 | + | 0.187268i | 0.940023 | + | 0.890437i | 4.37246 | − | 5.14766i | 3.52126 | + | 1.86685i | −0.447582 | − | 0.269301i | −2.61371 | − | 5.64945i | 2.92986 | + | 0.644911i | 13.7309 | − | 7.27968i |
10.17 | 2.78888 | + | 1.11119i | 1.72190 | + | 0.187268i | 3.63913 | + | 3.44717i | −3.64648 | + | 4.29297i | 4.59408 | + | 2.43563i | 3.88004 | + | 2.33454i | 1.27644 | + | 2.75897i | 2.92986 | + | 0.644911i | −14.9399 | + | 7.92064i |
10.18 | 3.16149 | + | 1.25965i | −1.72190 | − | 0.187268i | 5.50432 | + | 5.21397i | 3.15169 | − | 3.71045i | −5.20787 | − | 2.76104i | 6.22205 | + | 3.74368i | 5.11821 | + | 11.0628i | 2.92986 | + | 0.644911i | 14.6379 | − | 7.76053i |
10.19 | 3.27139 | + | 1.30344i | −1.72190 | − | 0.187268i | 6.09906 | + | 5.77734i | −5.84142 | + | 6.87706i | −5.38891 | − | 2.85702i | −0.280020 | − | 0.168482i | 6.50743 | + | 14.0656i | 2.92986 | + | 0.644911i | −28.0734 | + | 14.8836i |
10.20 | 3.40493 | + | 1.35665i | 1.72190 | + | 0.187268i | 6.84907 | + | 6.48778i | 1.18743 | − | 1.39795i | 5.60888 | + | 2.97364i | −5.85901 | − | 3.52525i | 8.36297 | + | 18.0763i | 2.92986 | + | 0.644911i | 5.93964 | − | 3.14900i |
See next 80 embeddings (of 560 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.d | odd | 58 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.3.g.a | ✓ | 560 |
59.d | odd | 58 | 1 | inner | 177.3.g.a | ✓ | 560 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.3.g.a | ✓ | 560 | 1.a | even | 1 | 1 | trivial |
177.3.g.a | ✓ | 560 | 59.d | odd | 58 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).