# Properties

 Label 429.2.bq.a Level $429$ Weight $2$ Character orbit 429.bq 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.bq (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$416$$ Relative dimension: $$52$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$416q - 3q^{3} + 42q^{4} - 5q^{6} - 10q^{7} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$416q - 3q^{3} + 42q^{4} - 5q^{6} - 10q^{7} + 3q^{9} - 52q^{12} - 20q^{13} - 3q^{15} + 30q^{16} - 20q^{18} - 10q^{19} + 4q^{22} + 5q^{24} + 60q^{25} - 24q^{27} - 10q^{28} - 65q^{30} - 8q^{31} + 49q^{33} - 24q^{34} + 3q^{36} - 6q^{37} - 35q^{39} - 120q^{40} - 13q^{42} - 32q^{45} - 30q^{46} - 33q^{48} - 26q^{49} - 100q^{51} - 60q^{52} - 42q^{55} - 20q^{57} - 34q^{58} + 32q^{60} - 30q^{61} - 45q^{63} + 40q^{64} + 42q^{66} + 32q^{67} - 33q^{69} - 268q^{70} + 55q^{72} - 40q^{73} + 9q^{75} - 32q^{78} - 37q^{81} + 28q^{82} - 180q^{84} - 10q^{85} - 142q^{88} - 30q^{90} + 8q^{91} - 26q^{93} - 90q^{94} + 330q^{96} - 14q^{97} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.276807 + 2.63364i −1.73182 0.0281544i −4.90316 1.04220i −1.71868 2.36556i 0.553529 4.55321i −0.0575585 + 0.270792i 2.46537 7.58762i 2.99841 + 0.0975169i 6.70579 3.87159i
29.2 −0.274469 + 2.61140i 1.03166 1.39129i −4.78776 1.01767i 1.02930 + 1.41670i 3.35004 + 3.07594i 0.916640 4.31245i 2.34881 7.22888i −0.871350 2.87067i −3.98208 + 2.29906i
29.3 −0.270097 + 2.56980i 0.0599513 1.73101i −4.57463 0.972368i −0.821874 1.13121i 4.43217 + 0.621604i −0.850348 + 4.00057i 2.13742 6.57829i −2.99281 0.207553i 3.12898 1.80652i
29.4 −0.260719 + 2.48057i 1.72227 + 0.183844i −4.12897 0.877640i −2.22258 3.05912i −0.905066 + 4.22428i 0.668489 3.14500i 1.71203 5.26908i 2.93240 + 0.633258i 8.16784 4.71570i
29.5 −0.256618 + 2.44156i 1.38266 + 1.04320i −3.93907 0.837274i 1.76595 + 2.43063i −2.90184 + 3.10814i −0.156026 + 0.734044i 1.53781 4.73290i 0.823485 + 2.88477i −6.38769 + 3.68794i
29.6 −0.244335 + 2.32469i −1.09963 + 1.33821i −3.38819 0.720181i 0.672844 + 0.926090i −2.84225 2.88327i −0.747086 + 3.51476i 1.05740 3.25434i −0.581631 2.94308i −2.31727 + 1.33788i
29.7 −0.237841 + 2.26290i −0.837076 1.51635i −3.10787 0.660597i 0.619671 + 0.852904i 3.63043 1.53357i 0.272540 1.28220i 0.827791 2.54768i −1.59861 + 2.53859i −2.07742 + 1.19940i
29.8 −0.230857 + 2.19646i −1.65359 0.515394i −2.81485 0.598316i 1.87210 + 2.57672i 1.51379 3.51307i −0.269880 + 1.26968i 0.599043 1.84366i 2.46874 + 1.70450i −6.09186 + 3.51714i
29.9 −0.219545 + 2.08883i 1.04164 + 1.38383i −2.35871 0.501359i −0.372847 0.513179i −3.11927 + 1.87199i −0.107574 + 0.506098i 0.267016 0.821790i −0.829974 + 2.88291i 1.15380 0.666147i
29.10 −0.198720 + 1.89069i −1.52365 + 0.823714i −1.57894 0.335614i −0.924919 1.27304i −1.25461 3.04444i 0.668009 3.14273i −0.226639 + 0.697522i 1.64299 2.51010i 2.59073 1.49576i
29.11 −0.188305 + 1.79160i 1.68236 0.411909i −1.21807 0.258909i −1.00123 1.37808i 0.421180 + 3.09168i −0.457516 + 2.15245i −0.420139 + 1.29305i 2.66066 1.38596i 2.65749 1.53430i
29.12 −0.179632 + 1.70909i 1.33049 1.10896i −0.932422 0.198192i 1.96134 + 2.69955i 1.65631 + 2.47313i −0.393158 + 1.84966i −0.555871 + 1.71080i 0.540418 2.95092i −4.96609 + 2.86717i
29.13 −0.170529 + 1.62248i −0.280130 + 1.70925i −0.647056 0.137536i −2.20891 3.04030i −2.72544 0.745981i −0.757508 + 3.56379i −0.674778 + 2.07675i −2.84305 0.957623i 5.30950 3.06544i
29.14 −0.159737 + 1.51979i −1.11400 1.32627i −0.327964 0.0697108i −0.0308880 0.0425136i 2.19361 1.48120i 0.593356 2.79152i −0.786124 + 2.41944i −0.517989 + 2.95494i 0.0695459 0.0401523i
29.15 −0.144098 + 1.37101i 0.143318 + 1.72611i 0.0974043 + 0.0207039i 0.756574 + 1.04133i −2.38716 0.0522407i 0.619507 2.91455i −0.894416 + 2.75273i −2.95892 + 0.494764i −1.53670 + 0.887212i
29.16 −0.127461 + 1.21271i 0.478976 1.66451i 0.501877 + 0.106677i −1.76681 2.43180i 1.95751 + 0.793018i 0.706704 3.32478i −0.946962 + 2.91445i −2.54116 1.59452i 3.17427 1.83266i
29.17 −0.123261 + 1.17275i −1.50937 0.849596i 0.596149 + 0.126715i −2.37562 3.26976i 1.18241 1.66539i −0.489968 + 2.30512i −0.950878 + 2.92650i 1.55637 + 2.56470i 4.12743 2.38297i
29.18 −0.115554 + 1.09942i 1.72615 0.142897i 0.760914 + 0.161737i 1.07486 + 1.47941i −0.0423587 + 1.91428i 0.576184 2.71073i −0.948969 + 2.92063i 2.95916 0.493323i −1.75071 + 1.01077i
29.19 −0.0983746 + 0.935971i −0.0424277 + 1.73153i 1.08993 + 0.231672i 1.83190 + 2.52139i −1.61649 0.210050i −0.116038 + 0.545914i −0.905708 + 2.78748i −2.99640 0.146930i −2.54016 + 1.46656i
29.20 −0.0790544 + 0.752152i −1.44283 + 0.958255i 1.39681 + 0.296902i 0.221314 + 0.304612i −0.606692 1.16098i 0.0924831 0.435099i −0.801155 + 2.46570i 1.16349 2.76519i −0.246611 + 0.142381i
See next 80 embeddings (of 416 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 425.52 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.d odd 10 1 inner
13.c even 3 1 inner
33.f even 10 1 inner
39.i odd 6 1 inner
143.t odd 30 1 inner
429.bq even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bq.a 416
3.b odd 2 1 inner 429.2.bq.a 416
11.d odd 10 1 inner 429.2.bq.a 416
13.c even 3 1 inner 429.2.bq.a 416
33.f even 10 1 inner 429.2.bq.a 416
39.i odd 6 1 inner 429.2.bq.a 416
143.t odd 30 1 inner 429.2.bq.a 416
429.bq even 30 1 inner 429.2.bq.a 416

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bq.a 416 1.a even 1 1 trivial
429.2.bq.a 416 3.b odd 2 1 inner
429.2.bq.a 416 11.d odd 10 1 inner
429.2.bq.a 416 13.c even 3 1 inner
429.2.bq.a 416 33.f even 10 1 inner
429.2.bq.a 416 39.i odd 6 1 inner
429.2.bq.a 416 143.t odd 30 1 inner
429.2.bq.a 416 429.bq even 30 1 inner

## Hecke kernels

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