↑

Properties

Label 229.2.i.a
Level $229$
Weight $2$
Character orbit 229.i
Analytic conductor $1.829$
Analytic rank $0$
Dimension $648$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [229,2,Mod(3,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(114))
 
chi = DirichletCharacter(H, H._module([104]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 229.i (of order \(57\), 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: \(1.82857420629\)
Analytic rank: \(0\)
Dimension: \(648\)
Relative dimension: \(18\) over \(\Q(\zeta_{57})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{57}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 32 q^{2} - 37 q^{3} - 60 q^{4} - 22 q^{5} - 43 q^{6} - 34 q^{7} - 20 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 648 q - 32 q^{2} - 37 q^{3} - 60 q^{4} - 22 q^{5} - 43 q^{6} - 34 q^{7} - 20 q^{8} - 25 q^{9} - 36 q^{10} - 30 q^{11} - 30 q^{12} + 26 q^{13} - 33 q^{14} - 42 q^{15} - 192 q^{16} - 32 q^{17} + 71 q^{18} - 34 q^{19} - 37 q^{20} - 68 q^{21} - 26 q^{22} + 42 q^{23} - 37 q^{24} - 8 q^{25} + 68 q^{26} - 40 q^{27} - 32 q^{28} - 61 q^{29} + 157 q^{30} - 183 q^{31} - 6 q^{32} - 75 q^{33} - 40 q^{34} - 15 q^{35} - 4 q^{36} - 36 q^{37} - 50 q^{38} - 31 q^{39} + 144 q^{40} - 53 q^{41} + 168 q^{42} + 22 q^{43} + 67 q^{44} - 38 q^{45} - 103 q^{46} - 37 q^{47} - 331 q^{48} + 58 q^{49} - 30 q^{50} - 51 q^{51} - 40 q^{52} + 113 q^{54} - 35 q^{55} + 198 q^{56} + 128 q^{57} - 28 q^{58} - q^{59} - 12 q^{60} - 278 q^{61} + 60 q^{62} - 190 q^{63} + 8 q^{64} - 20 q^{65} + 232 q^{66} + 4 q^{67} + 76 q^{68} + 63 q^{69} - 75 q^{70} + 43 q^{71} + 47 q^{72} - 25 q^{73} + 290 q^{74} + 73 q^{75} - 321 q^{76} - 51 q^{77} - 500 q^{78} + 9 q^{79} + 334 q^{80} + 88 q^{81} + 119 q^{82} + 5 q^{83} + 150 q^{84} + 86 q^{85} - 30 q^{86} - 67 q^{87} - 16 q^{88} - 31 q^{89} + 181 q^{90} - 267 q^{91} - 71 q^{92} - 107 q^{93} + 117 q^{94} + 176 q^{95} + 288 q^{96} - 17 q^{97} - 2 q^{98} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.43113 − 1.31566i 0.216111 − 0.210236i 3.08552 + 4.72273i −0.0837170 − 0.267658i −0.801992 + 0.226782i −0.112551 + 0.300611i −0.831220 − 10.0313i −0.0801582 + 2.90799i −0.148620 + 0.760853i
3.2 −1.95916 − 1.06025i 2.32105 − 2.25795i 1.62030 + 2.48005i 0.532040 + 1.70103i −6.94130 + 1.96281i 0.541504 − 1.44629i −0.177047 − 2.13664i 0.206262 − 7.48279i 0.761153 − 3.89668i
3.3 −1.84173 − 0.996695i 0.0872647 − 0.0848924i 1.30468 + 1.99696i 0.0910157 + 0.290993i −0.245330 + 0.0693728i 0.622382 − 1.66231i −0.0666442 − 0.804276i −0.0822546 + 2.98404i 0.122405 − 0.626647i
3.4 −1.73403 − 0.938408i −1.48293 + 1.44262i 1.03234 + 1.58011i −0.550732 − 1.76079i 3.92521 − 1.10994i −0.217416 + 0.580692i 0.0183246 + 0.221145i 0.0352761 − 1.27975i −0.697353 + 3.57006i
3.5 −1.66366 − 0.900327i −1.93619 + 1.88356i 0.863274 + 1.32134i 1.17902 + 3.76953i 4.91698 − 1.39039i 1.03212 − 2.75666i 0.0658670 + 0.794897i 0.118390 − 4.29499i 1.43233 − 7.33272i
3.6 −1.04420 − 0.565091i −1.03894 + 1.01070i −0.322878 − 0.494202i −0.968566 − 3.09668i 1.65599 − 0.468270i −0.474049 + 1.26613i 0.253972 + 3.06498i −0.0247739 + 0.898752i −0.738531 + 3.78087i
3.7 −1.03693 − 0.561159i 0.592960 − 0.576840i −0.333570 − 0.510567i 0.751404 + 2.40237i −0.938557 + 0.265399i −1.34129 + 3.58244i 0.254108 + 3.06663i −0.0638063 + 2.31477i 0.568959 − 2.91275i
3.8 −0.660253 − 0.357311i 1.55135 − 1.50917i −0.785634 − 1.20250i −0.581687 − 1.85976i −1.56353 + 0.442123i 0.430671 − 1.15027i 0.213040 + 2.57101i 0.0464095 − 1.68365i −0.280451 + 1.43575i
3.9 −0.00467277 − 0.00252878i −2.21436 + 2.15416i −1.09388 − 1.67431i −0.123603 − 0.395180i 0.0157946 − 0.00446628i 0.0623981 − 0.166658i 0.00175501 + 0.0211798i 0.180310 − 6.54131i −0.000421755 0.00215915i
3.10 0.269395 + 0.145789i −1.04199 + 1.01367i −1.04258 − 1.59578i 0.659264 + 2.10779i −0.428490 + 0.121165i −0.629065 + 1.68016i −0.0988074 − 1.19243i −0.0244332 + 0.886390i −0.129690 + 0.663941i
3.11 0.430112 + 0.232765i 1.53490 − 1.49318i −0.963079 − 1.47410i 1.10597 + 3.53597i 1.00774 − 0.284962i 0.980669 − 2.61925i −0.151885 − 1.83297i 0.0436878 − 1.58491i −0.347361 + 1.77830i
3.12 0.735589 + 0.398081i 0.874197 − 0.850432i −0.711273 − 1.08868i −0.617768 − 1.97511i 0.981591 − 0.277567i −0.370756 + 0.990246i −0.227959 − 2.75105i −0.0416772 + 1.51197i 0.331832 − 1.69880i
3.13 0.859606 + 0.465196i −0.907179 + 0.882517i −0.571381 − 0.874563i −0.742137 − 2.37274i −1.19036 + 0.336602i 1.71971 − 4.59314i −0.245747 − 2.96573i −0.0385262 + 1.39766i 0.465844 − 2.38486i
3.14 1.40353 + 0.759552i 2.33053 − 2.26718i 0.299079 + 0.457774i 0.117267 + 0.374923i 4.99301 − 1.41189i −1.75376 + 4.68410i −0.191510 − 2.31118i 0.208627 − 7.56859i −0.120186 + 0.615285i
3.15 1.63667 + 0.885721i −0.341121 + 0.331848i 0.800286 + 1.22493i 0.789257 + 2.52339i −0.852227 + 0.240987i 0.450042 − 1.20201i −0.0824962 − 0.995581i −0.0764223 + 2.77246i −0.943271 + 4.82902i
3.16 1.81078 + 0.979946i −0.938583 + 0.913068i 1.22474 + 1.87460i 0.0761271 + 0.243392i −2.59433 + 0.733606i −1.15795 + 3.09275i 0.0406742 + 0.490865i −0.0354175 + 1.28488i −0.100661 + 0.515330i
3.17 2.26915 + 1.22800i 0.603165 − 0.586768i 2.54715 + 3.89871i −1.23589 − 3.95137i 2.08922 − 0.590776i −0.106929 + 0.285595i 0.566119 + 6.83204i −0.0631517 + 2.29103i 2.04786 − 10.4839i
3.18 2.29345 + 1.24115i −2.15148 + 2.09299i 2.62555 + 4.01871i 0.171975 + 0.549834i −7.53202 + 2.12985i 1.41648 − 3.78326i 0.603049 + 7.27772i 0.165586 − 6.00717i −0.288012 + 1.47446i
9.1 −1.49954 − 2.29521i 0.0540931 − 1.96240i −2.21600 + 5.05196i −2.29838 + 1.59366i −4.58523 + 2.81853i −1.91256 − 1.66565i 9.50979 − 1.58690i −0.852629 − 0.0470409i 7.10429 + 2.88551i
9.2 −1.41055 − 2.15901i 0.00605773 − 0.219763i −1.86828 + 4.25925i 2.78091 − 1.92824i −0.483016 + 0.296909i 2.30759 + 2.00968i 6.74352 − 1.12529i 2.94719 + 0.162601i −8.08571 − 3.28413i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
229.i even 57 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.2.i.a ✓ 648
229.i even 57 1 inner 229.2.i.a ✓ 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.2.i.a ✓ 648 1.a even 1 1 trivial
229.2.i.a ✓ 648 229.i even 57 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(229, [\chi])\).