Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [229,3,Mod(2,229)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(76))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("229.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 229 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 229.j (of order \(76\), degree \(36\), 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: | \(6.23979805385\) |
Analytic rank: | \(0\) |
Dimension: | \(1368\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{76})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{76}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.80098 | + | 0.796982i | −4.17334 | − | 2.25850i | 10.1492 | − | 4.45184i | 8.06314 | + | 0.668131i | 17.6628 | + | 5.25844i | 7.10784 | − | 3.47481i | −22.3860 | + | 15.9833i | 7.39342 | + | 11.3165i | −31.1803 | + | 3.88662i |
2.2 | −3.61504 | + | 0.757994i | −1.32705 | − | 0.718166i | 8.83087 | − | 3.87358i | −5.61066 | − | 0.464912i | 5.34172 | + | 1.59030i | 0.486753 | − | 0.237959i | −16.9635 | + | 12.1117i | −3.67722 | − | 5.62841i | 20.6351 | − | 2.57217i |
2.3 | −3.55893 | + | 0.746229i | 4.18703 | + | 2.26591i | 8.44604 | − | 3.70478i | −1.04723 | − | 0.0867756i | −16.5923 | − | 4.93973i | 6.67354 | − | 3.26250i | −15.4566 | + | 11.0358i | 7.47438 | + | 11.4404i | 3.79176 | − | 0.472642i |
2.4 | −3.53314 | + | 0.740823i | 2.71340 | + | 1.46842i | 8.27120 | − | 3.62808i | 7.69168 | + | 0.637351i | −10.6747 | − | 3.17798i | −11.5224 | + | 5.63294i | −14.7837 | + | 10.5553i | 0.283741 | + | 0.434297i | −27.6480 | + | 3.44632i |
2.5 | −3.29926 | + | 0.691781i | 0.124122 | + | 0.0671715i | 6.74343 | − | 2.95794i | −0.362856 | − | 0.0300671i | −0.455978 | − | 0.135751i | −0.903136 | + | 0.441516i | −9.22812 | + | 6.58875i | −4.91164 | − | 7.51782i | 1.21795 | − | 0.151818i |
2.6 | −2.99165 | + | 0.627283i | −4.16513 | − | 2.25406i | 4.89338 | − | 2.14644i | −1.25098 | − | 0.103659i | 13.8745 | + | 4.13063i | −9.36336 | + | 4.57747i | −3.34208 | + | 2.38620i | 7.34501 | + | 11.2424i | 3.80751 | − | 0.474605i |
2.7 | −2.55131 | + | 0.534954i | 0.264748 | + | 0.143274i | 2.55992 | − | 1.12288i | 6.99679 | + | 0.579770i | −0.752098 | − | 0.223909i | 7.17138 | − | 3.50587i | 2.55568 | − | 1.82472i | −4.87297 | − | 7.45864i | −18.1611 | + | 2.26378i |
2.8 | −2.36818 | + | 0.496555i | 2.36169 | + | 1.27808i | 1.69862 | − | 0.745083i | −5.20631 | − | 0.431407i | −6.22755 | − | 1.85402i | −9.80223 | + | 4.79202i | 4.22435 | − | 3.01613i | −0.978451 | − | 1.49763i | 12.5437 | − | 1.56357i |
2.9 | −2.34414 | + | 0.491514i | −2.77460 | − | 1.50154i | 1.59031 | − | 0.697575i | 1.79372 | + | 0.148632i | 7.24209 | + | 2.15606i | −2.52727 | + | 1.23551i | 4.41200 | − | 3.15011i | 0.521268 | + | 0.797860i | −4.27779 | + | 0.533226i |
2.10 | −2.33517 | + | 0.489633i | 4.12933 | + | 2.23468i | 1.55017 | − | 0.679969i | −4.82452 | − | 0.399771i | −10.7368 | − | 3.19650i | 0.995453 | − | 0.486647i | 4.48022 | − | 3.19882i | 7.13502 | + | 10.9210i | 11.4618 | − | 1.42871i |
2.11 | −2.16437 | + | 0.453820i | −0.420617 | − | 0.227626i | 0.815431 | − | 0.357681i | −8.60573 | − | 0.713091i | 1.01367 | + | 0.301782i | 7.80875 | − | 3.81746i | 5.59652 | − | 3.99584i | −4.79743 | − | 7.34301i | 18.9496 | − | 2.36206i |
2.12 | −2.06087 | + | 0.432119i | −4.39642 | − | 2.37922i | 0.397366 | − | 0.174301i | −1.98839 | − | 0.164762i | 10.0886 | + | 3.00350i | 8.91982 | − | 4.36063i | 6.11124 | − | 4.36334i | 8.74529 | + | 13.3857i | 4.16900 | − | 0.519666i |
2.13 | −1.88662 | + | 0.395583i | 4.10205 | + | 2.21992i | −0.260232 | + | 0.114148i | 6.90878 | + | 0.572478i | −8.61719 | − | 2.56545i | −1.88661 | + | 0.922309i | 6.72107 | − | 4.79875i | 6.97625 | + | 10.6779i | −13.2607 | + | 1.65295i |
2.14 | −1.38122 | + | 0.289612i | −1.57523 | − | 0.852473i | −1.83919 | + | 0.806742i | 4.85842 | + | 0.402580i | 2.42263 | + | 0.721250i | −3.61313 | + | 1.76635i | 6.90090 | − | 4.92715i | −3.16789 | − | 4.84882i | −6.82716 | + | 0.851004i |
2.15 | −0.844161 | + | 0.177002i | 3.34040 | + | 1.80774i | −2.98182 | + | 1.30795i | 1.78615 | + | 0.148004i | −3.13981 | − | 0.934762i | 9.80847 | − | 4.79507i | 5.09346 | − | 3.63666i | 2.96785 | + | 4.54264i | −1.53399 | + | 0.191212i |
2.16 | −0.723194 | + | 0.151638i | 1.12055 | + | 0.606409i | −3.16308 | + | 1.38745i | 1.80041 | + | 0.149186i | −0.902326 | − | 0.268634i | −4.91157 | + | 2.40112i | 4.48261 | − | 3.20052i | −4.03464 | − | 6.17548i | −1.32467 | + | 0.165120i |
2.17 | −0.643155 | + | 0.134856i | −2.93887 | − | 1.59044i | −3.26763 | + | 1.43332i | −7.64386 | − | 0.633388i | 2.10463 | + | 0.626576i | −9.28248 | + | 4.53793i | 4.04756 | − | 2.88990i | 1.18495 | + | 1.81371i | 5.00161 | − | 0.623450i |
2.18 | −0.219716 | + | 0.0460696i | 1.17662 | + | 0.636756i | −3.61694 | + | 1.58654i | −4.48868 | − | 0.371943i | −0.287858 | − | 0.0856990i | 5.65778 | − | 2.76592i | 1.45243 | − | 1.03701i | −3.94355 | − | 6.03606i | 1.00337 | − | 0.125070i |
2.19 | −0.181862 | + | 0.0381325i | −4.58731 | − | 2.48253i | −3.63147 | + | 1.59291i | 9.27215 | + | 0.768312i | 0.928924 | + | 0.276553i | −4.35005 | + | 2.12661i | 1.20459 | − | 0.860064i | 9.95793 | + | 15.2417i | −1.71555 | + | 0.213843i |
2.20 | 0.110668 | − | 0.0232047i | −3.79642 | − | 2.05452i | −3.65138 | + | 1.60165i | −5.20436 | − | 0.431246i | −0.467817 | − | 0.139275i | 1.12513 | − | 0.550045i | −0.735029 | + | 0.524801i | 5.26921 | + | 8.06512i | −0.585964 | + | 0.0730403i |
See next 80 embeddings (of 1368 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
229.j | odd | 76 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 229.3.j.a | ✓ | 1368 |
229.j | odd | 76 | 1 | inner | 229.3.j.a | ✓ | 1368 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
229.3.j.a | ✓ | 1368 | 1.a | even | 1 | 1 | trivial |
229.3.j.a | ✓ | 1368 | 229.j | odd | 76 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(229, [\chi])\).