# Properties

 Label 429.2.bi.a Level $429$ Weight $2$ Character orbit 429.bi Analytic conductor $3.426$ Analytic rank $0$ Dimension $416$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bi (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$416$$ Relative dimension: $$52$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$416q - 12q^{3} - 14q^{6} - 12q^{7} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$416q - 12q^{3} - 14q^{6} - 12q^{7} - 12q^{9} - 20q^{13} - 30q^{15} + 32q^{16} + 2q^{18} - 4q^{19} - 12q^{21} - 24q^{22} - 78q^{24} - 36q^{27} - 84q^{28} - 28q^{31} - 44q^{33} - 24q^{34} - 12q^{37} + 54q^{39} + 88q^{40} - 56q^{42} + 8q^{45} - 92q^{46} + 40q^{48} - 44q^{52} - 176q^{54} - 72q^{55} - 6q^{57} - 4q^{58} + 12q^{60} - 48q^{61} - 46q^{63} + 204q^{66} - 64q^{67} + 56q^{70} - 66q^{72} - 12q^{73} - 104q^{76} - 92q^{78} + 104q^{79} + 124q^{81} + 16q^{84} - 12q^{85} - 24q^{87} - 84q^{91} - 124q^{93} + 328q^{94} - 152q^{96} + 52q^{97} - 142q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.23921 + 2.43208i 0.596820 1.62598i −3.20380 4.40966i −2.41132 + 1.22863i 3.21492 + 3.46643i 3.29586 + 0.522013i 9.30284 1.47343i −2.28761 1.94083i 7.38704i
5.2 −1.21529 + 2.38515i 1.68212 0.412863i −3.03642 4.17927i 2.86072 1.45761i −1.05954 + 4.51387i −3.01745 0.477918i 8.37041 1.32574i 2.65909 1.38897i 8.59467i
5.3 −1.17839 + 2.31273i −1.66274 + 0.485085i −2.78453 3.83258i −1.47438 + 0.751236i 0.837488 4.41708i 0.301846 + 0.0478077i 7.01764 1.11149i 2.52938 1.61314i 4.29510i
5.4 −1.15405 + 2.26495i −0.969560 + 1.43525i −2.62261 3.60971i 2.77016 1.41147i −2.13186 3.85237i 1.58890 + 0.251657i 6.18102 0.978978i −1.11991 2.78313i 7.90318i
5.5 −1.12326 + 2.20452i −0.671689 1.59651i −2.42261 3.33444i 0.958973 0.488621i 4.27400 + 0.312539i −2.80706 0.444595i 5.18460 0.821160i −2.09767 + 2.14471i 2.66292i
5.6 −1.04506 + 2.05104i 1.56544 + 0.741218i −1.93906 2.66889i 0.172910 0.0881019i −3.15625 + 2.43617i 3.48662 + 0.552227i 2.95325 0.467749i 1.90119 + 2.32066i 0.446717i
5.7 −1.02970 + 2.02089i 0.194927 + 1.72105i −1.84816 2.54377i 0.495829 0.252638i −3.67876 1.37823i −3.71395 0.588231i 2.56337 0.405999i −2.92401 + 0.670957i 1.26216i
5.8 −0.942586 + 1.84993i −1.52624 0.818902i −1.35820 1.86940i −2.61854 + 1.33421i 2.95352 2.05155i −3.28014 0.519524i 0.637161 0.100916i 1.65880 + 2.49968i 6.10173i
5.9 −0.881321 + 1.72969i 0.252737 1.71351i −1.03953 1.43079i 3.05267 1.55541i 2.74110 + 1.94731i 1.80249 + 0.285486i −0.443769 + 0.0702861i −2.87225 0.866135i 6.65099i
5.10 −0.838827 + 1.64629i 1.70467 + 0.306759i −0.831074 1.14388i −1.38189 + 0.704110i −1.93494 + 2.54907i −0.769893 0.121939i −1.06958 + 0.169404i 2.81180 + 1.04585i 2.86563i
5.11 −0.816343 + 1.60216i −0.787192 1.54283i −0.724943 0.997799i −0.438293 + 0.223321i 3.11449 0.00172985i 3.92191 + 0.621170i −1.36159 + 0.215654i −1.76066 + 2.42901i 0.884524i
5.12 −0.804126 + 1.57819i 1.16294 1.28358i −0.668483 0.920087i −3.07456 + 1.56656i 1.09058 + 2.86749i −2.02034 0.319990i −1.50925 + 0.239042i −0.295157 2.98545i 6.11194i
5.13 −0.698951 + 1.37177i −1.70131 + 0.324864i −0.217646 0.299564i −0.540754 + 0.275528i 0.743496 2.56087i 4.34543 + 0.688249i −2.47818 + 0.392505i 2.78893 1.10539i 0.934371i
5.14 −0.683621 + 1.34168i −1.71144 + 0.266392i −0.157203 0.216371i 3.31336 1.68824i 0.812565 2.47832i −3.03316 0.480405i −2.57676 + 0.408119i 2.85807 0.911829i 5.59959i
5.15 −0.667273 + 1.30960i 0.852386 + 1.50779i −0.0942200 0.129683i 3.33356 1.69853i −2.54337 + 0.110172i 0.619167 + 0.0980665i −2.67070 + 0.422997i −1.54688 + 2.57044i 5.49900i
5.16 −0.655660 + 1.28681i −0.748233 + 1.56210i −0.0504074 0.0693798i −2.04378 + 1.04136i −1.51953 1.98704i −1.36274 0.215838i −2.73054 + 0.432475i −1.88029 2.33763i 3.31273i
5.17 −0.507638 + 0.996296i 1.52061 0.829312i 0.440661 + 0.606518i 2.02501 1.03179i 0.0543225 + 1.93596i 2.69211 + 0.426388i −3.03677 + 0.480978i 1.62448 2.52211i 2.54129i
5.18 −0.482212 + 0.946394i −1.19043 + 1.25813i 0.512438 + 0.705310i −0.107143 + 0.0545921i −0.616644 1.73330i −1.02889 0.162960i −3.01277 + 0.477177i −0.165763 2.99542i 0.127724i
5.19 −0.339713 + 0.666725i −1.42645 0.982471i 0.846454 + 1.16504i 0.476840 0.242962i 1.13962 0.617289i −1.13337 0.179507i −2.54245 + 0.402685i 1.06950 + 2.80289i 0.400459i
5.20 −0.319309 + 0.626680i 1.28422 + 1.16223i 0.884801 + 1.21782i −2.81651 + 1.43508i −1.13841 + 0.433681i −1.31181 0.207770i −2.43507 + 0.385677i 0.298429 + 2.98512i 2.22328i
See next 80 embeddings (of 416 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 356.52 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.c even 5 1 inner
13.d odd 4 1 inner
33.h odd 10 1 inner
39.f even 4 1 inner
143.r odd 20 1 inner
429.bi even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bi.a 416
3.b odd 2 1 inner 429.2.bi.a 416
11.c even 5 1 inner 429.2.bi.a 416
13.d odd 4 1 inner 429.2.bi.a 416
33.h odd 10 1 inner 429.2.bi.a 416
39.f even 4 1 inner 429.2.bi.a 416
143.r odd 20 1 inner 429.2.bi.a 416
429.bi even 20 1 inner 429.2.bi.a 416

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bi.a 416 1.a even 1 1 trivial
429.2.bi.a 416 3.b odd 2 1 inner
429.2.bi.a 416 11.c even 5 1 inner
429.2.bi.a 416 13.d odd 4 1 inner
429.2.bi.a 416 33.h odd 10 1 inner
429.2.bi.a 416 39.f even 4 1 inner
429.2.bi.a 416 143.r odd 20 1 inner
429.2.bi.a 416 429.bi even 20 1 inner

## Hecke kernels

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