# Properties

 Label 429.2.bj.a Level $429$ Weight $2$ Character orbit 429.bj Analytic conductor $3.426$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$112q - 28q^{3} - 6q^{5} - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q - 28q^{3} - 6q^{5} - 28q^{9} - 10q^{11} + 10q^{13} - 60q^{14} + 4q^{15} + 80q^{16} - 74q^{20} + 8q^{22} + 30q^{24} + 38q^{26} - 28q^{27} - 20q^{29} - 8q^{31} + 20q^{33} - 48q^{34} + 20q^{35} + 12q^{37} - 10q^{39} + 40q^{40} - 110q^{41} + 20q^{42} - 36q^{44} + 4q^{45} - 20q^{46} - 30q^{47} - 20q^{48} + 90q^{50} - 10q^{52} + 52q^{53} - 64q^{55} + 30q^{57} + 24q^{58} - 36q^{59} - 74q^{60} - 60q^{61} + 48q^{66} + 60q^{67} + 60q^{68} + 116q^{70} + 20q^{71} - 30q^{72} + 70q^{73} + 120q^{74} - 52q^{78} - 120q^{79} + 8q^{80} - 28q^{81} + 30q^{83} + 30q^{84} - 40q^{85} + 62q^{86} + 48q^{89} - 4q^{91} - 144q^{92} - 8q^{93} - 20q^{94} + 82q^{97} - 10q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 −2.64667 + 0.419191i 0.309017 0.951057i 4.92702 1.60089i −2.31578 0.366783i −0.419191 + 2.64667i −2.21285 + 4.34297i −7.59393 + 3.86930i −0.809017 0.587785i 6.28285
73.2 −2.45153 + 0.388283i 0.309017 0.951057i 3.95710 1.28574i −1.53570 0.243230i −0.388283 + 2.45153i 2.15383 4.22713i −4.77859 + 2.43481i −0.809017 0.587785i 3.85924
73.3 −1.67413 + 0.265156i 0.309017 0.951057i 0.830297 0.269780i 1.80923 + 0.286554i −0.265156 + 1.67413i 0.0352382 0.0691588i 1.70202 0.867223i −0.809017 0.587785i −3.10487
73.4 −1.65467 + 0.262074i 0.309017 0.951057i 0.767143 0.249260i −1.95063 0.308949i −0.262074 + 1.65467i 1.17205 2.30028i 1.78136 0.907647i −0.809017 0.587785i 3.30862
73.5 −0.962531 + 0.152450i 0.309017 0.951057i −0.998888 + 0.324558i −3.13516 0.496561i −0.152450 + 0.962531i −0.833499 + 1.63583i 2.64861 1.34953i −0.809017 0.587785i 3.09339
73.6 −0.904052 + 0.143188i 0.309017 0.951057i −1.10531 + 0.359135i 1.60359 + 0.253983i −0.143188 + 0.904052i −2.17730 + 4.27319i 2.57894 1.31404i −0.809017 0.587785i −1.48609
73.7 −0.447102 + 0.0708140i 0.309017 0.951057i −1.70723 + 0.554712i 4.30790 + 0.682304i −0.0708140 + 0.447102i 1.76348 3.46103i 1.53070 0.779929i −0.809017 0.587785i −1.97439
73.8 0.502609 0.0796054i 0.309017 0.951057i −1.65583 + 0.538013i 0.387478 + 0.0613705i 0.0796054 0.502609i −1.68058 + 3.29833i −1.69623 + 0.864271i −0.809017 0.587785i 0.199635
73.9 0.523380 0.0828953i 0.309017 0.951057i −1.63506 + 0.531262i 1.16191 + 0.184028i 0.0828953 0.523380i 0.580635 1.13956i −1.75601 + 0.894734i −0.809017 0.587785i 0.623374
73.10 0.724135 0.114692i 0.309017 0.951057i −1.39090 + 0.451929i −1.84470 0.292172i 0.114692 0.724135i 1.08006 2.11974i −2.26187 + 1.15248i −0.809017 0.587785i −1.36932
73.11 1.92680 0.305175i 0.309017 0.951057i 1.71732 0.557991i 3.27451 + 0.518632i 0.305175 1.92680i −0.858865 + 1.68562i −0.337739 + 0.172087i −0.809017 0.587785i 6.46761
73.12 2.08536 0.330289i 0.309017 0.951057i 2.33753 0.759508i −3.67776 0.582500i 0.330289 2.08536i 1.72106 3.37776i 0.861266 0.438837i −0.809017 0.587785i −7.86185
73.13 2.19389 0.347478i 0.309017 0.951057i 2.79029 0.906622i 1.48353 + 0.234969i 0.347478 2.19389i 0.326561 0.640911i 1.84829 0.941752i −0.809017 0.587785i 3.33636
73.14 2.78451 0.441023i 0.309017 0.951057i 5.65687 1.83803i −1.82850 0.289605i 0.441023 2.78451i −1.06982 + 2.09963i 9.91711 5.05302i −0.809017 0.587785i −5.21918
112.1 −1.13354 2.22470i −0.809017 + 0.587785i −2.48880 + 3.42554i −0.0990937 + 0.194482i 2.22470 + 1.13354i −0.728003 4.59643i 5.50974 + 0.872657i 0.309017 0.951057i 0.544991
112.2 −1.01823 1.99839i −0.809017 + 0.587785i −1.78119 + 2.45160i −0.716768 + 1.40674i 1.99839 + 1.01823i 0.347529 + 2.19421i 2.28246 + 0.361506i 0.309017 0.951057i 3.54104
112.3 −0.753510 1.47885i −0.809017 + 0.587785i −0.443641 + 0.610619i 1.42413 2.79501i 1.47885 + 0.753510i 0.104512 + 0.659862i −2.04133 0.323315i 0.309017 0.951057i −5.20649
112.4 −0.661175 1.29763i −0.809017 + 0.587785i −0.0711199 + 0.0978882i −1.28004 + 2.51223i 1.29763 + 0.661175i −0.219997 1.38901i −2.70282 0.428085i 0.309017 0.951057i 4.10627
112.5 −0.501725 0.984691i −0.809017 + 0.587785i 0.457683 0.629946i 1.47255 2.89004i 0.984691 + 0.501725i −0.482591 3.04696i −3.03301 0.480381i 0.309017 0.951057i −3.58461
112.6 −0.379529 0.744868i −0.809017 + 0.587785i 0.764785 1.05264i −0.448929 + 0.881074i 0.744868 + 0.379529i 0.542880 + 3.42761i −2.72572 0.431711i 0.309017 0.951057i 0.826665
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.14 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.d odd 10 1 inner
13.d odd 4 1 inner
143.s 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.bj.a 112
11.d odd 10 1 inner 429.2.bj.a 112
13.d odd 4 1 inner 429.2.bj.a 112
143.s even 20 1 inner 429.2.bj.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bj.a 112 1.a even 1 1 trivial
429.2.bj.a 112 11.d odd 10 1 inner
429.2.bj.a 112 13.d odd 4 1 inner
429.2.bj.a 112 143.s even 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$21\!\cdots\!50$$$$T_{2}^{81} +$$$$19\!\cdots\!09$$$$T_{2}^{80} +$$$$17\!\cdots\!20$$$$T_{2}^{79} +$$$$40\!\cdots\!50$$$$T_{2}^{78} -$$$$50\!\cdots\!20$$$$T_{2}^{77} -$$$$34\!\cdots\!55$$$$T_{2}^{76} -$$$$34\!\cdots\!30$$$$T_{2}^{75} -$$$$10\!\cdots\!50$$$$T_{2}^{74} +$$$$80\!\cdots\!70$$$$T_{2}^{73} +$$$$51\!\cdots\!38$$$$T_{2}^{72} +$$$$64\!\cdots\!10$$$$T_{2}^{71} +$$$$24\!\cdots\!00$$$$T_{2}^{70} -$$$$13\!\cdots\!70$$$$T_{2}^{69} -$$$$66\!\cdots\!78$$$$T_{2}^{68} -$$$$92\!\cdots\!70$$$$T_{2}^{67} -$$$$34\!\cdots\!50$$$$T_{2}^{66} +$$$$18\!\cdots\!80$$$$T_{2}^{65} +$$$$74\!\cdots\!01$$$$T_{2}^{64} +$$$$10\!\cdots\!80$$$$T_{2}^{63} +$$$$63\!\cdots\!00$$$$T_{2}^{62} -$$$$12\!\cdots\!50$$$$T_{2}^{61} -$$$$56\!\cdots\!71$$$$T_{2}^{60} -$$$$86\!\cdots\!80$$$$T_{2}^{59} -$$$$71\!\cdots\!00$$$$T_{2}^{58} +$$$$13\!\cdots\!30$$$$T_{2}^{57} +$$$$24\!\cdots\!63$$$$T_{2}^{56} +$$$$45\!\cdots\!70$$$$T_{2}^{55} +$$$$44\!\cdots\!50$$$$T_{2}^{54} +$$$$10\!\cdots\!80$$$$T_{2}^{53} -$$$$62\!\cdots\!17$$$$T_{2}^{52} -$$$$10\!\cdots\!70$$$$T_{2}^{51} -$$$$39\!\cdots\!00$$$$T_{2}^{50} +$$$$24\!\cdots\!00$$$$T_{2}^{49} +$$$$86\!\cdots\!40$$$$T_{2}^{48} +$$$$14\!\cdots\!00$$$$T_{2}^{47} +$$$$17\!\cdots\!00$$$$T_{2}^{46} +$$$$16\!\cdots\!80$$$$T_{2}^{45} +$$$$11\!\cdots\!63$$$$T_{2}^{44} +$$$$64\!\cdots\!30$$$$T_{2}^{43} +$$$$10\!\cdots\!50$$$$T_{2}^{42} +$$$$32\!\cdots\!10$$$$T_{2}^{41} +$$$$73\!\cdots\!96$$$$T_{2}^{40} +$$$$11\!\cdots\!90$$$$T_{2}^{39} +$$$$12\!\cdots\!00$$$$T_{2}^{38} +$$$$96\!\cdots\!20$$$$T_{2}^{37} +$$$$29\!\cdots\!43$$$$T_{2}^{36} -$$$$45\!\cdots\!50$$$$T_{2}^{35} -$$$$88\!\cdots\!00$$$$T_{2}^{34} -$$$$84\!\cdots\!80$$$$T_{2}^{33} -$$$$45\!\cdots\!77$$$$T_{2}^{32} +$$$$18\!\cdots\!90$$$$T_{2}^{31} +$$$$29\!\cdots\!50$$$$T_{2}^{30} +$$$$31\!\cdots\!00$$$$T_{2}^{29} +$$$$17\!\cdots\!48$$$$T_{2}^{28} +$$$$17\!\cdots\!70$$$$T_{2}^{27} -$$$$55\!\cdots\!50$$$$T_{2}^{26} -$$$$55\!\cdots\!00$$$$T_{2}^{25} -$$$$24\!\cdots\!32$$$$T_{2}^{24} -$$$$27\!\cdots\!30$$$$T_{2}^{23} +$$$$61\!\cdots\!50$$$$T_{2}^{22} +$$$$45\!\cdots\!50$$$$T_{2}^{21} +$$$$16\!\cdots\!45$$$$T_{2}^{20} -$$$$13\!\cdots\!80$$$$T_{2}^{19} -$$$$28\!\cdots\!50$$$$T_{2}^{18} -$$$$20\!\cdots\!50$$$$T_{2}^{17} -$$$$69\!\cdots\!12$$$$T_{2}^{16} +$$$$41\!\cdots\!90$$$$T_{2}^{15} +$$$$97\!\cdots\!00$$$$T_{2}^{14} +$$$$72\!\cdots\!40$$$$T_{2}^{13} +$$$$26\!\cdots\!31$$$$T_{2}^{12} -$$$$21\!\cdots\!90$$$$T_{2}^{11} -$$$$36\!\cdots\!50$$$$T_{2}^{10} -$$$$25\!\cdots\!70$$$$T_{2}^{9} -$$$$88\!\cdots\!05$$$$T_{2}^{8} +$$$$12\!\cdots\!20$$$$T_{2}^{7} +$$$$10\!\cdots\!50$$$$T_{2}^{6} +$$$$71\!\cdots\!80$$$$T_{2}^{5} +$$$$22\!\cdots\!31$$$$T_{2}^{4} -$$$$37\!\cdots\!50$$$$T_{2}^{3} + 274184935200 T_{2}^{2} - 10841953320 T_{2} + 214358881$$">$$T_{2}^{112} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.