Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,2,Mod(29,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 9, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 208.bj (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: | \(1.66088836204\) |
Analytic rank: | \(0\) |
Dimension: | \(104\) |
Relative dimension: | \(26\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −1.41044 | + | 0.103276i | −0.623697 | + | 2.32767i | 1.97867 | − | 0.291328i | −2.33843 | + | 2.33843i | 0.639294 | − | 3.34744i | −3.16277 | − | 1.82602i | −2.76070 | + | 0.615248i | −2.43097 | − | 1.40352i | 3.05670 | − | 3.53971i |
29.2 | −1.39507 | + | 0.231899i | 0.0845055 | − | 0.315379i | 1.89245 | − | 0.647031i | 2.63109 | − | 2.63109i | −0.0447552 | + | 0.459573i | −3.34168 | − | 1.92932i | −2.49005 | + | 1.34151i | 2.50575 | + | 1.44670i | −3.06041 | + | 4.28070i |
29.3 | −1.39445 | − | 0.235600i | 0.156066 | − | 0.582446i | 1.88899 | + | 0.657065i | −0.921546 | + | 0.921546i | −0.354850 | + | 0.775422i | 1.44176 | + | 0.832400i | −2.47929 | − | 1.36129i | 2.28319 | + | 1.31820i | 1.50217 | − | 1.06793i |
29.4 | −1.34109 | − | 0.448861i | −0.762293 | + | 2.84492i | 1.59705 | + | 1.20393i | 2.03642 | − | 2.03642i | 2.29928 | − | 3.47313i | 3.62835 | + | 2.09483i | −1.60139 | − | 2.33143i | −4.91439 | − | 2.83732i | −3.64509 | + | 1.81695i |
29.5 | −1.28175 | + | 0.597591i | 0.722632 | − | 2.69690i | 1.28577 | − | 1.53193i | 0.609482 | − | 0.609482i | 0.685410 | + | 3.88859i | 3.93274 | + | 2.27057i | −0.732570 | + | 2.73191i | −4.15300 | − | 2.39773i | −0.416983 | + | 1.14542i |
29.6 | −0.979428 | − | 1.02016i | 0.598940 | − | 2.23527i | −0.0814433 | + | 1.99834i | 2.39927 | − | 2.39927i | −2.86695 | + | 1.57828i | −0.442240 | − | 0.255327i | 2.11839 | − | 1.87415i | −2.03964 | − | 1.17759i | −4.79754 | − | 0.0977225i |
29.7 | −0.967244 | + | 1.03172i | 0.371119 | − | 1.38503i | −0.128880 | − | 1.99584i | −2.99717 | + | 2.99717i | 1.07000 | + | 1.72255i | −1.68939 | − | 0.975368i | 2.18380 | + | 1.79750i | 0.817487 | + | 0.471976i | −0.193237 | − | 5.99123i |
29.8 | −0.890578 | − | 1.09858i | 0.100590 | − | 0.375407i | −0.413740 | + | 1.95674i | −1.52134 | + | 1.52134i | −0.501996 | + | 0.223823i | 0.798611 | + | 0.461078i | 2.51809 | − | 1.28810i | 2.46726 | + | 1.42448i | 3.02617 | + | 0.316435i |
29.9 | −0.851616 | − | 1.12905i | −0.520271 | + | 1.94168i | −0.549502 | + | 1.92303i | −0.499375 | + | 0.499375i | 2.63532 | − | 1.06615i | −2.35238 | − | 1.35814i | 2.63916 | − | 1.01727i | −0.901359 | − | 0.520400i | 0.989095 | + | 0.138543i |
29.10 | −0.672039 | + | 1.24433i | 0.0577913 | − | 0.215680i | −1.09673 | − | 1.67248i | 0.668900 | − | 0.668900i | 0.229540 | + | 0.216857i | −0.616743 | − | 0.356077i | 2.81816 | − | 0.240721i | 2.55490 | + | 1.47507i | 0.382807 | + | 1.28186i |
29.11 | −0.237188 | + | 1.39418i | −0.623936 | + | 2.32856i | −1.88748 | − | 0.661365i | −1.66259 | + | 1.66259i | −3.09845 | − | 1.42219i | 2.78550 | + | 1.60821i | 1.36975 | − | 2.47463i | −2.43482 | − | 1.40575i | −1.92361 | − | 2.71230i |
29.12 | −0.176994 | − | 1.40309i | −0.465351 | + | 1.73671i | −1.93735 | + | 0.496679i | 1.82441 | − | 1.82441i | 2.51914 | + | 0.345544i | 0.405225 | + | 0.233957i | 1.03979 | + | 2.63037i | −0.201550 | − | 0.116365i | −2.88273 | − | 2.23691i |
29.13 | −0.0791950 | + | 1.41199i | 0.815998 | − | 3.04535i | −1.98746 | − | 0.223646i | 0.626691 | − | 0.626691i | 4.23539 | + | 1.39336i | −1.94682 | − | 1.12400i | 0.473183 | − | 2.78857i | −6.01021 | − | 3.47000i | 0.835254 | + | 0.934515i |
29.14 | −0.0263250 | − | 1.41397i | 0.465695 | − | 1.73800i | −1.99861 | + | 0.0744453i | −0.627944 | + | 0.627944i | −2.46973 | − | 0.612725i | −4.03746 | − | 2.33103i | 0.157877 | + | 2.82402i | −0.205684 | − | 0.118752i | 0.904424 | + | 0.871363i |
29.15 | 0.266053 | + | 1.38896i | −0.0165565 | + | 0.0617898i | −1.85843 | + | 0.739075i | 2.60178 | − | 2.60178i | −0.0902286 | − | 0.00655704i | 2.68608 | + | 1.55081i | −1.52099 | − | 2.38466i | 2.59453 | + | 1.49795i | 4.30598 | + | 2.92156i |
29.16 | 0.298771 | − | 1.38229i | −0.354336 | + | 1.32240i | −1.82147 | − | 0.825978i | −2.17499 | + | 2.17499i | 1.72208 | + | 0.884891i | 3.13079 | + | 1.80756i | −1.68595 | + | 2.27103i | 0.974889 | + | 0.562852i | 2.35665 | + | 3.65630i |
29.17 | 0.443624 | − | 1.34283i | 0.502442 | − | 1.87514i | −1.60640 | − | 1.19142i | 1.07638 | − | 1.07638i | −2.29510 | − | 1.50655i | 2.22572 | + | 1.28502i | −2.31252 | + | 1.62858i | −0.665623 | − | 0.384298i | −0.967891 | − | 1.92291i |
29.18 | 0.586716 | + | 1.28676i | −0.356544 | + | 1.33064i | −1.31153 | + | 1.50993i | −1.01509 | + | 1.01509i | −1.92141 | + | 0.321920i | −3.02737 | − | 1.74785i | −2.71242 | − | 0.801727i | 0.954598 | + | 0.551137i | −1.90175 | − | 0.710612i |
29.19 | 0.884002 | + | 1.10388i | 0.301612 | − | 1.12563i | −0.437081 | + | 1.95166i | −2.02015 | + | 2.02015i | 1.50918 | − | 0.662118i | 2.08993 | + | 1.20662i | −2.54077 | + | 1.24278i | 1.42200 | + | 0.820993i | −4.01580 | − | 0.444176i |
29.20 | 0.995696 | − | 1.00429i | −0.356961 | + | 1.33220i | −0.0171782 | − | 1.99993i | 1.82137 | − | 1.82137i | 0.982481 | + | 1.68495i | −1.32362 | − | 0.764192i | −2.02560 | − | 1.97407i | 0.950750 | + | 0.548916i | −0.0156441 | − | 3.64270i |
See next 80 embeddings (of 104 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
16.e | even | 4 | 1 | inner |
208.bj | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 208.2.bj.a | ✓ | 104 |
4.b | odd | 2 | 1 | 832.2.br.a | 104 | ||
13.c | even | 3 | 1 | inner | 208.2.bj.a | ✓ | 104 |
16.e | even | 4 | 1 | inner | 208.2.bj.a | ✓ | 104 |
16.f | odd | 4 | 1 | 832.2.br.a | 104 | ||
52.j | odd | 6 | 1 | 832.2.br.a | 104 | ||
208.bg | odd | 12 | 1 | 832.2.br.a | 104 | ||
208.bj | even | 12 | 1 | inner | 208.2.bj.a | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
208.2.bj.a | ✓ | 104 | 1.a | even | 1 | 1 | trivial |
208.2.bj.a | ✓ | 104 | 13.c | even | 3 | 1 | inner |
208.2.bj.a | ✓ | 104 | 16.e | even | 4 | 1 | inner |
208.2.bj.a | ✓ | 104 | 208.bj | even | 12 | 1 | inner |
832.2.br.a | 104 | 4.b | odd | 2 | 1 | ||
832.2.br.a | 104 | 16.f | odd | 4 | 1 | ||
832.2.br.a | 104 | 52.j | odd | 6 | 1 | ||
832.2.br.a | 104 | 208.bg | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(208, [\chi])\).