# Properties

 Label 429.2.x.a Level $429$ Weight $2$ Character orbit 429.x Analytic conductor $3.426$ Analytic rank $0$ Dimension $192$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.x (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$48$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$192q + 6q^{3} - 48q^{4} - 10q^{6} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$192q + 6q^{3} - 48q^{4} - 10q^{6} - 14q^{9} - 20q^{12} + 6q^{15} - 28q^{16} - 10q^{18} - 60q^{19} + 4q^{22} + 30q^{24} - 12q^{25} + 30q^{27} - 20q^{28} + 12q^{31} - 32q^{33} + 56q^{34} + 44q^{36} + 60q^{40} - 108q^{42} - 40q^{45} - 28q^{48} - 24q^{49} + 10q^{51} + 20q^{52} - 20q^{55} + 50q^{57} - 60q^{58} + 36q^{60} + 20q^{61} - 30q^{63} + 64q^{64} - 30q^{66} - 120q^{67} - 16q^{69} + 20q^{70} + 150q^{72} - 20q^{73} + 18q^{75} + 40q^{78} - 160q^{79} - 46q^{81} + 88q^{82} - 40q^{85} + 72q^{88} - 80q^{90} + 8q^{91} - 72q^{93} - 120q^{94} - 140q^{96} - 88q^{97} + 62q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
248.1 −0.867138 2.66878i −1.64023 + 0.556462i −4.75241 + 3.45283i 1.12653 + 0.366031i 2.90738 + 3.89488i −1.16567 1.60441i 8.79543 + 6.39025i 2.38070 1.82545i 3.32385i
248.2 −0.833208 2.56435i 1.66893 + 0.463321i −4.26363 + 3.09771i −3.14288 1.02118i −0.202449 4.66577i −1.19340 1.64257i 7.13336 + 5.18269i 2.57067 + 1.54650i 8.91030i
248.3 −0.828482 2.54980i 1.34014 1.09728i −4.19709 + 3.04936i 2.56502 + 0.833425i −3.90813 2.50803i 2.78275 + 3.83012i 6.91450 + 5.02368i 0.591966 2.94102i 7.23077i
248.4 −0.759437 2.33731i 0.0396224 + 1.73160i −3.26823 + 2.37451i −0.541474 0.175936i 4.01719 1.40765i 0.184926 + 0.254529i 4.05551 + 2.94650i −2.99686 + 0.137220i 1.39920i
248.5 −0.751690 2.31346i −0.447888 1.67314i −3.16905 + 2.30245i 1.59703 + 0.518906i −3.53408 + 2.29386i −2.23366 3.07437i 3.77287 + 2.74115i −2.59879 + 1.49876i 4.08473i
248.6 −0.742222 2.28432i −0.941406 1.45388i −3.04921 + 2.21538i −1.40487 0.456471i −2.62239 + 3.22957i 2.72019 + 3.74402i 3.43751 + 2.49750i −1.22751 + 2.73737i 3.54799i
248.7 −0.653756 2.01206i 1.33811 + 1.09976i −2.00293 + 1.45522i 0.460929 + 0.149765i 1.33797 3.41132i 1.17111 + 1.61190i 0.814296 + 0.591620i 0.581071 + 2.94319i 1.02532i
248.8 −0.641276 1.97365i 1.56505 0.742034i −1.86601 + 1.35573i 2.08634 + 0.677894i −2.46814 2.61301i −1.98628 2.73388i 0.514597 + 0.373877i 1.89877 2.32264i 4.55242i
248.9 −0.639563 1.96837i −0.957560 + 1.44329i −1.84742 + 1.34223i 0.0542705 + 0.0176336i 3.45335 + 0.961761i 0.902973 + 1.24284i 0.474744 + 0.344922i −1.16616 2.76407i 0.118102i
248.10 −0.588772 1.81206i −1.41376 + 1.00065i −1.31886 + 0.958206i −4.08513 1.32734i 2.64561 + 1.97265i −0.445193 0.612756i −0.570025 0.414147i 0.997417 2.82934i 8.18397i
248.11 −0.585970 1.80343i 1.29904 1.14565i −1.29097 + 0.937942i −2.64888 0.860673i −2.82729 1.67141i −0.0612752 0.0843381i −0.620198 0.450600i 0.374988 2.97647i 5.28140i
248.12 −0.471771 1.45196i 0.741004 + 1.56554i −0.267592 + 0.194417i −0.890567 0.289363i 1.92352 1.81449i −2.98183 4.10413i −2.06170 1.49791i −1.90183 + 2.32014i 1.42958i
248.13 −0.403403 1.24155i 0.347020 1.69693i 0.239326 0.173881i −1.63352 0.530762i −2.24681 + 0.253706i 0.605010 + 0.832725i −2.42467 1.76163i −2.75915 1.17774i 2.24220i
248.14 −0.395367 1.21681i 1.57722 + 0.715804i 0.293714 0.213396i 3.00322 + 0.975804i 0.247420 2.20219i 0.904708 + 1.24522i −2.44595 1.77709i 1.97525 + 2.25796i 4.04015i
248.15 −0.387859 1.19371i −0.747764 + 1.56232i 0.343533 0.249591i 4.12008 + 1.33870i 2.15498 + 0.286650i −1.94633 2.67890i −2.46204 1.78877i −1.88170 2.33649i 5.43740i
248.16 −0.358717 1.10402i −0.371629 1.69171i 0.527855 0.383509i 2.25266 + 0.731932i −1.73437 + 1.01713i 0.626805 + 0.862723i −2.49102 1.80983i −2.72378 + 1.25738i 2.74953i
248.17 −0.316587 0.974355i −1.11557 1.32495i 0.768894 0.558634i −1.73318 0.563145i −0.937801 + 1.50642i −2.49336 3.43181i −2.44540 1.77669i −0.511010 + 2.95616i 1.86702i
248.18 −0.268430 0.826144i −1.73136 + 0.0488003i 1.00758 0.732046i −1.29465 0.420659i 0.505067 + 1.41726i 2.39963 + 3.30281i −2.28076 1.65707i 2.99524 0.168982i 1.18249i
248.19 −0.214393 0.659835i −0.480417 + 1.66409i 1.22862 0.892642i 1.23103 + 0.399986i 1.20102 0.0397745i 2.53769 + 3.49283i −1.97498 1.43491i −2.53840 1.59891i 0.898032i
248.20 −0.155057 0.477215i 1.66619 + 0.473072i 1.41434 1.02758i −2.24502 0.729452i −0.0325974 0.868487i −1.56756 2.15756i −1.52157 1.10548i 2.55241 + 1.57646i 1.18447i
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 404.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.x.a 192
3.b odd 2 1 inner 429.2.x.a 192
11.d odd 10 1 inner 429.2.x.a 192
33.f even 10 1 inner 429.2.x.a 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.x.a 192 1.a even 1 1 trivial
429.2.x.a 192 3.b odd 2 1 inner
429.2.x.a 192 11.d odd 10 1 inner
429.2.x.a 192 33.f even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(429, [\chi])$$.