Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [229,6,Mod(228,229)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("229.228");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 229 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 229.b (of order \(2\), degree \(1\), 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: | \(36.7278947372\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
228.1 | − | 11.1781i | −3.16796 | −92.9502 | −36.0444 | 35.4118i | − | 205.527i | 681.308i | −232.964 | 402.909i | ||||||||||||||||
228.2 | − | 11.0691i | 20.3502 | −90.5255 | −59.1732 | − | 225.259i | 131.149i | 647.826i | 171.132 | 654.995i | ||||||||||||||||
228.3 | − | 10.8821i | −24.9326 | −86.4201 | 65.6943 | 271.319i | 15.9659i | 592.204i | 378.632 | − | 714.892i | ||||||||||||||||
228.4 | − | 10.7844i | 23.6485 | −84.3038 | 102.758 | − | 255.036i | − | 182.411i | 564.067i | 316.252 | − | 1108.19i | ||||||||||||||
228.5 | − | 10.3591i | −6.41074 | −75.3115 | 51.7956 | 66.4097i | 123.693i | 448.669i | −201.902 | − | 536.557i | ||||||||||||||||
228.6 | − | 10.3382i | −19.0212 | −74.8785 | −50.7666 | 196.645i | 158.870i | 443.287i | 118.806 | 524.835i | |||||||||||||||||
228.7 | − | 10.0458i | 12.8746 | −68.9178 | 80.8957 | − | 129.335i | 242.245i | 370.868i | −77.2456 | − | 812.661i | |||||||||||||||
228.8 | − | 9.58045i | 8.28048 | −59.7850 | 20.6062 | − | 79.3307i | − | 10.4936i | 266.193i | −174.434 | − | 197.417i | ||||||||||||||
228.9 | − | 9.44857i | −0.232652 | −57.2755 | −60.8224 | 2.19823i | 25.1573i | 238.817i | −242.946 | 574.685i | |||||||||||||||||
228.10 | − | 9.44209i | 24.0152 | −57.1530 | −15.4633 | − | 226.754i | − | 54.0704i | 237.497i | 333.730 | 146.006i | |||||||||||||||
228.11 | − | 9.36871i | −25.1047 | −55.7727 | −71.2647 | 235.199i | − | 103.514i | 222.719i | 387.246 | 667.658i | ||||||||||||||||
228.12 | − | 9.31923i | −14.1475 | −54.8481 | 75.1824 | 131.844i | − | 25.9900i | 212.927i | −42.8488 | − | 700.642i | |||||||||||||||
228.13 | − | 9.29308i | −2.06582 | −54.3613 | 17.9634 | 19.1979i | − | 135.055i | 207.805i | −238.732 | − | 166.935i | |||||||||||||||
228.14 | − | 8.86792i | −21.5966 | −46.6401 | 57.8760 | 191.517i | − | 243.487i | 129.827i | 223.412 | − | 513.240i | |||||||||||||||
228.15 | − | 8.77076i | 21.6275 | −44.9263 | −82.5416 | − | 189.690i | − | 118.386i | 113.373i | 224.750 | 723.953i | |||||||||||||||
228.16 | − | 8.48650i | 0.511501 | −40.0207 | −96.1183 | − | 4.34085i | 113.816i | 68.0679i | −242.738 | 815.708i | ||||||||||||||||
228.17 | − | 7.95178i | 28.6379 | −31.2308 | 11.7195 | − | 227.723i | − | 26.4903i | − | 6.11615i | 577.131 | − | 93.1907i | |||||||||||||
228.18 | − | 7.73247i | 22.6779 | −27.7911 | 32.6209 | − | 175.356i | 158.597i | − | 32.5452i | 271.286 | − | 252.240i | ||||||||||||||
228.19 | − | 7.37565i | 6.73372 | −22.4002 | 86.4248 | − | 49.6656i | − | 55.5190i | − | 70.8044i | −197.657 | − | 637.439i | |||||||||||||
228.20 | − | 7.35415i | −15.0620 | −22.0835 | 20.6981 | 110.768i | 13.8828i | − | 72.9273i | −16.1350 | − | 152.217i | |||||||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
229.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 229.6.b.a | ✓ | 96 |
229.b | even | 2 | 1 | inner | 229.6.b.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
229.6.b.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
229.6.b.a | ✓ | 96 | 229.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(229, [\chi])\).