# Properties

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

## Newform invariants

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

 $$\operatorname{Tr}(f)(q) =$$ $$112q - 14q^{3} - 20q^{4} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q - 14q^{3} - 20q^{4} + 14q^{9} + 21q^{11} + 120q^{12} - 15q^{13} + 30q^{14} - 6q^{15} - 8q^{16} - 12q^{19} - 66q^{20} - 17q^{22} + 12q^{23} + 14q^{25} + 9q^{26} + 28q^{27} + 18q^{28} - 2q^{29} + 10q^{30} - 30q^{32} - 24q^{33} + 16q^{35} - 20q^{36} + 38q^{38} - 12q^{39} - 60q^{40} - 36q^{41} + 12q^{43} - 6q^{45} + 54q^{46} - 42q^{48} - 40q^{49} - 51q^{50} - 30q^{51} - 15q^{52} - 22q^{53} + 22q^{55} - 76q^{56} + 132q^{58} + 72q^{59} + 34q^{61} + 17q^{62} - 84q^{64} + 28q^{65} - 24q^{66} - 48q^{67} - 12q^{68} - 7q^{69} + 30q^{71} + 20q^{74} + 32q^{75} - 48q^{76} - 136q^{77} - 14q^{78} - 36q^{79} + 18q^{80} + 14q^{81} - 5q^{82} - 27q^{84} - 66q^{85} + 52q^{87} + 52q^{88} - 96q^{89} - 20q^{90} - 125q^{91} - 10q^{92} - 18q^{93} + 22q^{94} + 72q^{95} - 93q^{97} + 24q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −1.05844 2.37729i −0.669131 0.743145i −3.19297 + 3.54616i 0.696965 + 0.959290i −1.05844 + 2.37729i 1.37323 + 1.23646i 6.86002 + 2.22895i −0.104528 + 0.994522i 1.54282 2.67224i
4.2 −0.866606 1.94643i −0.669131 0.743145i −1.69932 + 1.88729i 1.80442 + 2.48357i −0.866606 + 1.94643i −3.61054 3.25094i 1.09341 + 0.355271i −0.104528 + 0.994522i 3.27037 5.66445i
4.3 −0.763963 1.71589i −0.669131 0.743145i −1.02237 + 1.13546i −1.07133 1.47456i −0.763963 + 1.71589i 1.14179 + 1.02807i −0.843308 0.274007i −0.104528 + 0.994522i −1.71172 + 2.96479i
4.4 −0.688193 1.54571i −0.669131 0.743145i −0.577339 + 0.641200i 1.57124 + 2.16263i −0.688193 + 1.54571i 1.25124 + 1.12663i −1.82992 0.594578i −0.104528 + 0.994522i 2.26148 3.91699i
4.5 −0.672174 1.50973i −0.669131 0.743145i −0.489197 + 0.543308i −2.52085 3.46966i −0.672174 + 1.50973i −2.16497 1.94935i −1.99436 0.648008i −0.104528 + 0.994522i −3.54378 + 6.13802i
4.6 −0.177986 0.399764i −0.669131 0.743145i 1.21013 1.34398i 0.0311778 + 0.0429126i −0.177986 + 0.399764i −0.860670 0.774951i −1.58502 0.515004i −0.104528 + 0.994522i 0.0116057 0.0201016i
4.7 −0.0803440 0.180456i −0.669131 0.743145i 1.31215 1.45729i 0.296705 + 0.408379i −0.0803440 + 0.180456i 3.31863 + 2.98811i −0.744131 0.241783i −0.104528 + 0.994522i 0.0498559 0.0863529i
4.8 0.250968 + 0.563684i −0.669131 0.743145i 1.08351 1.20336i −1.79971 2.47709i 0.250968 0.563684i −2.23596 2.01327i 2.12390 + 0.690096i −0.104528 + 0.994522i 0.944626 1.63614i
4.9 0.275867 + 0.619607i −0.669131 0.743145i 1.03045 1.14443i −1.77061 2.43703i 0.275867 0.619607i 3.22654 + 2.90519i 2.28346 + 0.741942i −0.104528 + 0.994522i 1.02155 1.76938i
4.10 0.386229 + 0.867484i −0.669131 0.743145i 0.734905 0.816195i 2.09746 + 2.88691i 0.386229 0.867484i 0.681566 + 0.613685i 2.79809 + 0.909153i −0.104528 + 0.994522i −1.69425 + 2.93453i
4.11 0.534181 + 1.19979i −0.669131 0.743145i 0.184114 0.204479i 1.16055 + 1.59736i 0.534181 1.19979i −2.62438 2.36301i 2.84179 + 0.923354i −0.104528 + 0.994522i −1.29655 + 2.24569i
4.12 0.883617 + 1.98464i −0.669131 0.743145i −1.81974 + 2.02103i −1.20025 1.65200i 0.883617 1.98464i −2.60068 2.34166i −1.48670 0.483058i −0.104528 + 0.994522i 2.21806 3.84180i
4.13 0.896442 + 2.01344i −0.669131 0.743145i −1.91208 + 2.12358i 0.00844073 + 0.0116177i 0.896442 2.01344i 1.50438 + 1.35455i −1.79754 0.584056i −0.104528 + 0.994522i −0.0158249 + 0.0274095i
4.14 1.08040 + 2.42662i −0.669131 0.743145i −3.38297 + 3.75717i 2.34307 + 3.22496i 1.08040 2.42662i 1.59982 + 1.44048i −7.71968 2.50828i −0.104528 + 0.994522i −5.29430 + 9.17000i
49.1 −1.97287 1.77638i 0.104528 0.994522i 0.527630 + 5.02007i 1.64057 + 0.533054i −1.97287 + 1.77638i −2.55155 + 0.268179i 4.75574 6.54571i −0.978148 0.207912i −2.28973 3.96592i
49.2 −1.80408 1.62440i 0.104528 0.994522i 0.406968 + 3.87204i −2.99760 0.973979i −1.80408 + 1.62440i 1.24847 0.131220i 2.70169 3.71856i −0.978148 0.207912i 3.82577 + 6.62643i
49.3 −1.54435 1.39054i 0.104528 0.994522i 0.242362 + 2.30592i 1.99851 + 0.649356i −1.54435 + 1.39054i 2.22364 0.233714i 0.389193 0.535678i −0.978148 0.207912i −2.18345 3.78184i
49.4 −1.38170 1.24409i 0.104528 0.994522i 0.152285 + 1.44889i −1.73448 0.563568i −1.38170 + 1.24409i −4.66758 + 0.490582i −0.593557 + 0.816960i −0.978148 0.207912i 1.69541 + 2.93654i
49.5 −1.04336 0.939450i 0.104528 0.994522i −0.00301284 0.0286652i 1.68655 + 0.547992i −1.04336 + 0.939450i −0.0876448 + 0.00921184i −1.67427 + 2.30444i −0.978148 0.207912i −1.24487 2.15618i
49.6 −0.280336 0.252415i 0.104528 0.994522i −0.194182 1.84752i −2.49942 0.812112i −0.280336 + 0.252415i −1.73582 + 0.182442i −0.855366 + 1.17731i −0.978148 0.207912i 0.495688 + 0.858557i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 400.14 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.c even 5 1 inner
13.e even 6 1 inner
143.u 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.bn.a 112
11.c even 5 1 inner 429.2.bn.a 112
13.e even 6 1 inner 429.2.bn.a 112
143.u even 30 1 inner 429.2.bn.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bn.a 112 1.a even 1 1 trivial
429.2.bn.a 112 11.c even 5 1 inner
429.2.bn.a 112 13.e even 6 1 inner
429.2.bn.a 112 143.u even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$25\!\cdots\!34$$$$T_{2}^{84} - 603643065171 T_{2}^{83} -$$$$14\!\cdots\!86$$$$T_{2}^{82} +$$$$10\!\cdots\!60$$$$T_{2}^{81} +$$$$53\!\cdots\!19$$$$T_{2}^{80} +$$$$78\!\cdots\!51$$$$T_{2}^{79} +$$$$32\!\cdots\!46$$$$T_{2}^{78} +$$$$18\!\cdots\!57$$$$T_{2}^{77} +$$$$73\!\cdots\!97$$$$T_{2}^{76} -$$$$51\!\cdots\!20$$$$T_{2}^{75} -$$$$15\!\cdots\!67$$$$T_{2}^{74} -$$$$49\!\cdots\!52$$$$T_{2}^{73} -$$$$18\!\cdots\!08$$$$T_{2}^{72} -$$$$15\!\cdots\!88$$$$T_{2}^{71} -$$$$74\!\cdots\!69$$$$T_{2}^{70} -$$$$17\!\cdots\!66$$$$T_{2}^{69} -$$$$12\!\cdots\!08$$$$T_{2}^{68} +$$$$54\!\cdots\!09$$$$T_{2}^{67} +$$$$35\!\cdots\!09$$$$T_{2}^{66} +$$$$40\!\cdots\!16$$$$T_{2}^{65} +$$$$30\!\cdots\!93$$$$T_{2}^{64} +$$$$14\!\cdots\!81$$$$T_{2}^{63} +$$$$98\!\cdots\!59$$$$T_{2}^{62} +$$$$43\!\cdots\!35$$$$T_{2}^{61} +$$$$15\!\cdots\!50$$$$T_{2}^{60} +$$$$17\!\cdots\!02$$$$T_{2}^{59} -$$$$13\!\cdots\!76$$$$T_{2}^{58} +$$$$65\!\cdots\!77$$$$T_{2}^{57} -$$$$14\!\cdots\!12$$$$T_{2}^{56} +$$$$13\!\cdots\!97$$$$T_{2}^{55} -$$$$34\!\cdots\!25$$$$T_{2}^{54} -$$$$34\!\cdots\!83$$$$T_{2}^{53} +$$$$45\!\cdots\!53$$$$T_{2}^{52} -$$$$12\!\cdots\!80$$$$T_{2}^{51} +$$$$22\!\cdots\!76$$$$T_{2}^{50} -$$$$39\!\cdots\!43$$$$T_{2}^{49} +$$$$56\!\cdots\!95$$$$T_{2}^{48} -$$$$45\!\cdots\!42$$$$T_{2}^{47} +$$$$28\!\cdots\!57$$$$T_{2}^{46} +$$$$75\!\cdots\!11$$$$T_{2}^{45} -$$$$18\!\cdots\!95$$$$T_{2}^{44} +$$$$40\!\cdots\!19$$$$T_{2}^{43} -$$$$60\!\cdots\!16$$$$T_{2}^{42} +$$$$79\!\cdots\!47$$$$T_{2}^{41} -$$$$88\!\cdots\!43$$$$T_{2}^{40} +$$$$61\!\cdots\!92$$$$T_{2}^{39} -$$$$28\!\cdots\!67$$$$T_{2}^{38} -$$$$13\!\cdots\!31$$$$T_{2}^{37} +$$$$31\!\cdots\!46$$$$T_{2}^{36} -$$$$53\!\cdots\!34$$$$T_{2}^{35} +$$$$73\!\cdots\!63$$$$T_{2}^{34} -$$$$83\!\cdots\!69$$$$T_{2}^{33} +$$$$80\!\cdots\!84$$$$T_{2}^{32} -$$$$63\!\cdots\!68$$$$T_{2}^{31} +$$$$41\!\cdots\!91$$$$T_{2}^{30} -$$$$20\!\cdots\!32$$$$T_{2}^{29} +$$$$71\!\cdots\!47$$$$T_{2}^{28} -$$$$86\!\cdots\!39$$$$T_{2}^{27} -$$$$10\!\cdots\!05$$$$T_{2}^{26} +$$$$10\!\cdots\!36$$$$T_{2}^{25} -$$$$51\!\cdots\!75$$$$T_{2}^{24} +$$$$18\!\cdots\!68$$$$T_{2}^{23} -$$$$42\!\cdots\!17$$$$T_{2}^{22} +$$$$85\!\cdots\!64$$$$T_{2}^{21} +$$$$12\!\cdots\!47$$$$T_{2}^{20} -$$$$26\!\cdots\!48$$$$T_{2}^{19} +$$$$14\!\cdots\!02$$$$T_{2}^{18} -$$$$58\!\cdots\!17$$$$T_{2}^{17} +$$$$65\!\cdots\!71$$$$T_{2}^{16} -$$$$17\!\cdots\!05$$$$T_{2}^{15} +$$$$37\!\cdots\!15$$$$T_{2}^{14} -$$$$11\!\cdots\!45$$$$T_{2}^{13} +$$$$71\!\cdots\!85$$$$T_{2}^{12} +$$$$43\!\cdots\!75$$$$T_{2}^{11} +$$$$11\!\cdots\!50$$$$T_{2}^{10} +$$$$27\!\cdots\!25$$$$T_{2}^{9} +$$$$37\!\cdots\!00$$$$T_{2}^{8} +$$$$58\!\cdots\!00$$$$T_{2}^{7} +$$$$10\!\cdots\!50$$$$T_{2}^{6} +$$$$16\!\cdots\!00$$$$T_{2}^{5} + 292490938375 T_{2}^{4} + 26808127500 T_{2}^{3} + 2455619375 T_{2}^{2} + 152233125 T_{2} + 9150625$$">$$T_{2}^{112} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.