Properties

Label 229.6.b.a
Level $229$
Weight $6$
Character orbit 229.b
Analytic conductor $36.728$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 18 q^{3} - 1590 q^{4} + 126 q^{5} + 7494 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 18 q^{3} - 1590 q^{4} + 126 q^{5} + 7494 q^{9} - 466 q^{11} - 1416 q^{12} - 474 q^{14} - 2550 q^{15} + 26354 q^{16} + 1844 q^{17} - 2660 q^{19} - 13428 q^{20} + 63498 q^{25} - 8856 q^{26} + 12036 q^{27} + 18556 q^{33} - 101192 q^{36} - 28610 q^{37} + 16850 q^{42} + 1300 q^{43} - 15974 q^{44} - 19456 q^{45} + 63460 q^{46} + 15040 q^{48} - 292740 q^{49} - 15958 q^{51} + 72128 q^{53} + 70432 q^{55} + 137206 q^{56} + 29796 q^{57} + 116196 q^{58} + 106456 q^{60} - 70868 q^{61} - 145664 q^{62} - 480106 q^{64} - 107048 q^{68} - 203648 q^{70} - 134202 q^{71} + 228800 q^{75} - 84630 q^{76} - 223286 q^{78} + 596050 q^{80} + 241952 q^{81} - 271664 q^{82} + 10070 q^{83} + 54546 q^{85} + 366456 q^{91} - 199882 q^{94} + 259574 q^{95} + 269688 q^{97} + 155766 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 228.96
Significant digits:
Format:

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 \(S_{6}^{\mathrm{new}}(229, [\chi])\).