# Properties

 Label 429.2.n.d Level $429$ Weight $2$ Character orbit 429.n Analytic conductor $3.426$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$9$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 3q^{2} - 9q^{3} - 11q^{4} + 3q^{6} + q^{7} - q^{8} - 9q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 3q^{2} - 9q^{3} - 11q^{4} + 3q^{6} + q^{7} - q^{8} - 9q^{9} + 6q^{10} - 10q^{11} + 54q^{12} + 9q^{13} - 5q^{14} - 10q^{15} - 13q^{16} - 2q^{18} + 10q^{19} + 37q^{20} - 14q^{21} - 9q^{22} + 18q^{23} + 4q^{24} - 31q^{25} + 2q^{26} - 9q^{27} + 12q^{28} + 10q^{29} + q^{30} - 28q^{31} - 74q^{32} + 5q^{33} + 40q^{34} - 14q^{35} - 11q^{36} - 26q^{37} + 7q^{38} + 9q^{39} - 72q^{40} + 26q^{41} - 5q^{42} + 4q^{43} - 68q^{44} + 20q^{45} - 57q^{46} - 28q^{48} - 18q^{49} + 11q^{50} - 5q^{51} + 11q^{52} + 11q^{53} - 2q^{54} - 32q^{55} + 72q^{56} + 50q^{58} + 55q^{59} + 37q^{60} + 14q^{61} - 50q^{62} + q^{63} - q^{64} - 20q^{65} - 14q^{66} + 104q^{67} - 9q^{68} + 8q^{69} + 44q^{70} - 8q^{71} + 4q^{72} - 3q^{73} + 69q^{74} - 21q^{75} - 52q^{76} + 2q^{77} + 2q^{78} - 19q^{79} - 159q^{80} - 9q^{81} + 58q^{82} + 12q^{83} - 8q^{84} + 63q^{86} - 97q^{88} + 118q^{89} - 4q^{90} - q^{91} + 98q^{92} - 28q^{93} - 99q^{94} - 45q^{95} + q^{96} + 50q^{97} - 186q^{98} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1 −2.09213 + 1.52002i 0.309017 + 0.951057i 1.44850 4.45804i 0.296923 + 0.215727i −2.09213 1.52002i 0.607096 1.86845i 2.14760 + 6.60965i −0.809017 + 0.587785i −0.949111
157.2 −1.61703 + 1.17484i 0.309017 + 0.951057i 0.616508 1.89742i 2.77955 + 2.01946i −1.61703 1.17484i −1.25981 + 3.87730i −0.00305069 0.00938907i −0.809017 + 0.587785i −6.86718
157.3 −1.57947 + 1.14755i 0.309017 + 0.951057i 0.559820 1.72295i −3.22160 2.34063i −1.57947 1.14755i 0.498644 1.53467i −0.113653 0.349788i −0.809017 + 0.587785i 7.77443
157.4 −0.214873 + 0.156114i 0.309017 + 0.951057i −0.596235 + 1.83502i −3.10912 2.25891i −0.214873 0.156114i −0.868527 + 2.67305i −0.322507 0.992575i −0.809017 + 0.587785i 1.02072
157.5 0.167740 0.121870i 0.309017 + 0.951057i −0.604750 + 1.86123i 0.762452 + 0.553954i 0.167740 + 0.121870i 0.163043 0.501795i 0.253530 + 0.780285i −0.809017 + 0.587785i 0.195404
157.6 1.19143 0.865625i 0.309017 + 0.951057i 0.0521663 0.160551i −0.917011 0.666248i 1.19143 + 0.865625i −1.26104 + 3.88108i 0.833347 + 2.56478i −0.809017 + 0.587785i −1.66928
157.7 1.30744 0.949910i 0.309017 + 0.951057i 0.189034 0.581786i 1.69854 + 1.23406i 1.30744 + 0.949910i 1.36281 4.19429i 0.693300 + 2.13376i −0.809017 + 0.587785i 3.39298
157.8 1.89306 1.37539i 0.309017 + 0.951057i 1.07394 3.30525i −2.78298 2.02196i 1.89306 + 1.37539i 0.597560 1.83910i −1.06680 3.28328i −0.809017 + 0.587785i −8.04932
157.9 2.25286 1.63680i 0.309017 + 0.951057i 1.77823 5.47283i 2.25719 + 1.63994i 2.25286 + 1.63680i −1.26682 + 3.89888i −3.23079 9.94334i −0.809017 + 0.587785i 7.76939
196.1 −0.839420 + 2.58347i −0.809017 + 0.587785i −4.35166 3.16166i 0.0697046 + 0.214529i −0.839420 2.58347i −1.77678 1.29090i 7.42567 5.39506i 0.309017 0.951057i −0.612740
196.2 −0.711780 + 2.19063i −0.809017 + 0.587785i −2.67421 1.94293i 0.686429 + 2.11261i −0.711780 2.19063i 3.55434 + 2.58238i 2.43276 1.76750i 0.309017 0.951057i −5.11655
196.3 −0.311986 + 0.960194i −0.809017 + 0.587785i 0.793397 + 0.576437i 1.17262 + 3.60896i −0.311986 0.960194i −4.12734 2.99869i −2.43460 + 1.76884i 0.309017 0.951057i −3.83114
196.4 −0.264345 + 0.813569i −0.809017 + 0.587785i 1.02602 + 0.745445i −0.893708 2.75055i −0.264345 0.813569i 3.14423 + 2.28442i −2.26182 + 1.64331i 0.309017 0.951057i 2.47401
196.5 −0.0340599 + 0.104826i −0.809017 + 0.587785i 1.60821 + 1.16843i 0.850571 + 2.61779i −0.0340599 0.104826i 2.32344 + 1.68808i −0.355597 + 0.258356i 0.309017 0.951057i −0.303382
196.6 0.396567 1.22051i −0.809017 + 0.587785i 0.285661 + 0.207545i 0.189502 + 0.583226i 0.396567 + 1.22051i −0.578859 0.420566i 2.44304 1.77498i 0.309017 0.951057i 0.786982
196.7 0.423381 1.30303i −0.809017 + 0.587785i 0.0993922 + 0.0722127i −1.34435 4.13749i 0.423381 + 1.30303i 0.830818 + 0.603625i 2.35303 1.70957i 0.309017 0.951057i −5.96045
196.8 0.693505 2.13439i −0.809017 + 0.587785i −2.45663 1.78484i 0.185200 + 0.569986i 0.693505 + 2.13439i −3.43059 2.49247i −1.88200 + 1.36735i 0.309017 0.951057i 1.34501
196.9 0.839121 2.58255i −0.809017 + 0.587785i −4.34740 3.15857i 1.32010 + 4.06285i 0.839121 + 2.58255i 1.98779 + 1.44422i −7.41147 + 5.38475i 0.309017 0.951057i 11.6002
235.1 −2.09213 1.52002i 0.309017 0.951057i 1.44850 + 4.45804i 0.296923 0.215727i −2.09213 + 1.52002i 0.607096 + 1.86845i 2.14760 6.60965i −0.809017 0.587785i −0.949111
235.2 −1.61703 1.17484i 0.309017 0.951057i 0.616508 + 1.89742i 2.77955 2.01946i −1.61703 + 1.17484i −1.25981 3.87730i −0.00305069 + 0.00938907i −0.809017 0.587785i −6.86718
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.d 36
11.c even 5 1 inner 429.2.n.d 36
11.c even 5 1 4719.2.a.bq 18
11.d odd 10 1 4719.2.a.br 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.d 36 1.a even 1 1 trivial
429.2.n.d 36 11.c even 5 1 inner
4719.2.a.bq 18 11.c even 5 1
4719.2.a.br 18 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.