# Properties

 Label 429.2.bs.b Level $429$ Weight $2$ Character orbit 429.bs Analytic conductor $3.426$ Analytic rank $0$ Dimension $224$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bs (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$14$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$224q + 28q^{3} + 6q^{5} + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$224q + 28q^{3} + 6q^{5} + 28q^{9} - 8q^{11} - 10q^{13} + 36q^{14} + 8q^{15} - 8q^{16} - 22q^{20} + 10q^{22} - 48q^{23} - 30q^{24} - 2q^{26} - 56q^{27} + 20q^{29} - 4q^{31} + 28q^{33} - 72q^{34} - 20q^{35} - 36q^{37} - 20q^{39} + 80q^{40} - 70q^{41} - 8q^{42} - 24q^{44} - 4q^{45} - 40q^{46} - 42q^{47} - 28q^{48} + 24q^{49} + 180q^{50} + 10q^{52} - 4q^{53} + 82q^{55} - 72q^{56} - 132q^{58} - 66q^{59} - 16q^{60} + 60q^{61} - 120q^{62} - 60q^{66} - 12q^{67} - 60q^{68} - 12q^{69} - 44q^{70} - 32q^{71} + 30q^{72} - 70q^{73} + 60q^{74} + 12q^{75} + 52q^{78} - 120q^{79} - 44q^{80} + 28q^{81} + 6q^{82} - 240q^{83} + 60q^{84} + 40q^{85} + 10q^{86} + 6q^{88} - 24q^{89} + 88q^{91} + 48q^{92} + 20q^{93} - 40q^{94} - 300q^{95} - 46q^{97} + 10q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −0.987158 + 2.57163i 0.669131 + 0.743145i −4.15253 3.73895i 3.32910 + 0.527277i −2.57163 + 0.987158i 0.866370 0.0454045i 8.80570 4.48673i −0.104528 + 0.994522i −4.64231 + 8.04071i
7.2 −0.893395 + 2.32737i 0.669131 + 0.743145i −3.13222 2.82026i −3.41301 0.540567i −2.32737 + 0.893395i 1.64776 0.0863554i 4.91964 2.50668i −0.104528 + 0.994522i 4.30726 7.46040i
7.3 −0.742514 + 1.93431i 0.669131 + 0.743145i −1.70395 1.53425i 0.399672 + 0.0633018i −1.93431 + 0.742514i −2.63195 + 0.137934i 0.540719 0.275510i −0.104528 + 0.994522i −0.419207 + 0.726088i
7.4 −0.473247 + 1.23285i 0.669131 + 0.743145i 0.190334 + 0.171377i 4.09856 + 0.649148i −1.23285 + 0.473247i −1.65326 + 0.0866439i −2.65461 + 1.35259i −0.104528 + 0.994522i −2.73993 + 4.74570i
7.5 −0.436789 + 1.13787i 0.669131 + 0.743145i 0.382318 + 0.344240i −0.0897712 0.0142184i −1.13787 + 0.436789i 4.09917 0.214828i −2.73066 + 1.39134i −0.104528 + 0.994522i 0.0553897 0.0959378i
7.6 −0.215576 + 0.561596i 0.669131 + 0.743145i 1.21737 + 1.09613i 0.0568875 + 0.00901009i −0.561596 + 0.215576i 3.00661 0.157570i −1.94999 + 0.993569i −0.104528 + 0.994522i −0.0173236 + 0.0300054i
7.7 −0.158810 + 0.413715i 0.669131 + 0.743145i 1.34035 + 1.20686i −3.17932 0.503554i −0.413715 + 0.158810i −0.944050 + 0.0494756i −1.50185 + 0.765232i −0.104528 + 0.994522i 0.713236 1.23536i
7.8 0.0361343 0.0941330i 0.669131 + 0.743145i 1.47873 + 1.33146i 2.04988 + 0.324669i 0.0941330 0.0361343i −4.15746 + 0.217883i 0.358448 0.182638i −0.104528 + 0.994522i 0.104633 0.181230i
7.9 0.342749 0.892893i 0.669131 + 0.743145i 0.806509 + 0.726184i 2.10352 + 0.333164i 0.892893 0.342749i 0.180391 0.00945392i 2.62918 1.33964i −0.104528 + 0.994522i 1.01846 1.76402i
7.10 0.400304 1.04283i 0.669131 + 0.743145i 0.559044 + 0.503365i −0.769642 0.121899i 1.04283 0.400304i 0.172980 0.00906547i 2.73925 1.39572i −0.104528 + 0.994522i −0.435211 + 0.753807i
7.11 0.727798 1.89598i 0.669131 + 0.743145i −1.57876 1.42152i 1.62284 + 0.257032i 1.89598 0.727798i 1.47389 0.0772434i −0.225156 + 0.114723i −0.104528 + 0.994522i 1.66843 2.88980i
7.12 0.750080 1.95402i 0.669131 + 0.743145i −1.76930 1.59309i −2.73934 0.433869i 1.95402 0.750080i 4.52723 0.237262i −0.710221 + 0.361876i −0.104528 + 0.994522i −2.90252 + 5.02731i
7.13 0.758057 1.97481i 0.669131 + 0.743145i −1.83892 1.65577i −3.65267 0.578526i 1.97481 0.758057i −4.58954 + 0.240527i −0.894330 + 0.455684i −0.104528 + 0.994522i −3.91141 + 6.77476i
7.14 0.892365 2.32469i 0.669131 + 0.743145i −3.12158 2.81068i 3.27062 + 0.518016i 2.32469 0.892365i −1.99815 + 0.104719i −4.88221 + 2.48761i −0.104528 + 0.994522i 4.12282 7.14093i
19.1 −1.56477 + 1.93233i −0.978148 0.207912i −0.869572 4.09102i −0.513743 + 0.0813689i 1.93233 1.56477i 1.92549 2.96500i 4.83500 + 2.46356i 0.913545 + 0.406737i 0.646659 1.12005i
19.2 −1.48670 + 1.83592i −0.978148 0.207912i −0.744504 3.50262i 0.0628305 0.00995138i 1.83592 1.48670i −0.583961 + 0.899221i 3.32757 + 1.69548i 0.913545 + 0.406737i −0.0751401 + 0.130146i
19.3 −1.36075 + 1.68039i −0.978148 0.207912i −0.556241 2.61691i 4.09781 0.649030i 1.68039 1.36075i −0.0869532 + 0.133896i 1.30116 + 0.662975i 0.913545 + 0.406737i −4.48550 + 7.76911i
19.4 −1.18625 + 1.46489i −0.978148 0.207912i −0.322905 1.51915i −2.00150 + 0.317006i 1.46489 1.18625i −1.72749 + 2.66010i −0.750595 0.382447i 0.913545 + 0.406737i 1.90989 3.30802i
19.5 −0.821551 + 1.01453i −0.978148 0.207912i 0.0614966 + 0.289319i −3.20835 + 0.508153i 1.01453 0.821551i −0.510870 + 0.786671i −2.67039 1.36063i 0.913545 + 0.406737i 2.12029 3.67244i
19.6 −0.560625 + 0.692314i −0.978148 0.207912i 0.250825 + 1.18004i 1.71423 0.271507i 0.692314 0.560625i 0.284275 0.437746i −2.54507 1.29678i 0.913545 + 0.406737i −0.773071 + 1.33900i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.14 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.f odd 12 1 inner
143.w even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bs.b 224
11.d odd 10 1 inner 429.2.bs.b 224
13.f odd 12 1 inner 429.2.bs.b 224
143.w even 60 1 inner 429.2.bs.b 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bs.b 224 1.a even 1 1 trivial
429.2.bs.b 224 11.d odd 10 1 inner
429.2.bs.b 224 13.f odd 12 1 inner
429.2.bs.b 224 143.w even 60 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!20$$$$T_{2}^{195} +$$$$63\!\cdots\!16$$$$T_{2}^{194} -$$$$18\!\cdots\!00$$$$T_{2}^{193} +$$$$58\!\cdots\!85$$$$T_{2}^{192} -$$$$74\!\cdots\!20$$$$T_{2}^{191} +$$$$18\!\cdots\!46$$$$T_{2}^{190} -$$$$56\!\cdots\!40$$$$T_{2}^{189} +$$$$25\!\cdots\!82$$$$T_{2}^{188} -$$$$18\!\cdots\!50$$$$T_{2}^{187} +$$$$13\!\cdots\!74$$$$T_{2}^{186} -$$$$39\!\cdots\!70$$$$T_{2}^{185} +$$$$40\!\cdots\!56$$$$T_{2}^{184} +$$$$20\!\cdots\!00$$$$T_{2}^{183} -$$$$26\!\cdots\!72$$$$T_{2}^{182} +$$$$85\!\cdots\!20$$$$T_{2}^{181} -$$$$13\!\cdots\!62$$$$T_{2}^{180} +$$$$28\!\cdots\!40$$$$T_{2}^{179} -$$$$11\!\cdots\!54$$$$T_{2}^{178} +$$$$38\!\cdots\!20$$$$T_{2}^{177} -$$$$98\!\cdots\!90$$$$T_{2}^{176} +$$$$11\!\cdots\!20$$$$T_{2}^{175} -$$$$24\!\cdots\!12$$$$T_{2}^{174} +$$$$70\!\cdots\!20$$$$T_{2}^{173} -$$$$27\!\cdots\!95$$$$T_{2}^{172} +$$$$21\!\cdots\!80$$$$T_{2}^{171} +$$$$56\!\cdots\!81$$$$T_{2}^{170} -$$$$84\!\cdots\!40$$$$T_{2}^{169} -$$$$14\!\cdots\!58$$$$T_{2}^{168} -$$$$34\!\cdots\!10$$$$T_{2}^{167} +$$$$24\!\cdots\!81$$$$T_{2}^{166} -$$$$97\!\cdots\!90$$$$T_{2}^{165} +$$$$19\!\cdots\!86$$$$T_{2}^{164} -$$$$40\!\cdots\!90$$$$T_{2}^{163} +$$$$10\!\cdots\!50$$$$T_{2}^{162} -$$$$31\!\cdots\!00$$$$T_{2}^{161} +$$$$76\!\cdots\!85$$$$T_{2}^{160} -$$$$11\!\cdots\!50$$$$T_{2}^{159} +$$$$19\!\cdots\!23$$$$T_{2}^{158} -$$$$31\!\cdots\!20$$$$T_{2}^{157} +$$$$90\!\cdots\!60$$$$T_{2}^{156} -$$$$11\!\cdots\!80$$$$T_{2}^{155} -$$$$16\!\cdots\!91$$$$T_{2}^{154} +$$$$10\!\cdots\!10$$$$T_{2}^{153} -$$$$17\!\cdots\!15$$$$T_{2}^{152} +$$$$66\!\cdots\!10$$$$T_{2}^{151} -$$$$15\!\cdots\!80$$$$T_{2}^{150} +$$$$44\!\cdots\!10$$$$T_{2}^{149} -$$$$73\!\cdots\!15$$$$T_{2}^{148} +$$$$16\!\cdots\!90$$$$T_{2}^{147} -$$$$24\!\cdots\!94$$$$T_{2}^{146} +$$$$59\!\cdots\!00$$$$T_{2}^{145} -$$$$73\!\cdots\!00$$$$T_{2}^{144} +$$$$10\!\cdots\!20$$$$T_{2}^{143} +$$$$49\!\cdots\!92$$$$T_{2}^{142} -$$$$15\!\cdots\!50$$$$T_{2}^{141} +$$$$82\!\cdots\!41$$$$T_{2}^{140} -$$$$23\!\cdots\!70$$$$T_{2}^{139} +$$$$76\!\cdots\!02$$$$T_{2}^{138} -$$$$16\!\cdots\!90$$$$T_{2}^{137} +$$$$32\!\cdots\!34$$$$T_{2}^{136} -$$$$61\!\cdots\!40$$$$T_{2}^{135} +$$$$13\!\cdots\!61$$$$T_{2}^{134} -$$$$26\!\cdots\!10$$$$T_{2}^{133} +$$$$49\!\cdots\!77$$$$T_{2}^{132} -$$$$87\!\cdots\!20$$$$T_{2}^{131} +$$$$15\!\cdots\!38$$$$T_{2}^{130} -$$$$29\!\cdots\!90$$$$T_{2}^{129} +$$$$57\!\cdots\!79$$$$T_{2}^{128} -$$$$10\!\cdots\!90$$$$T_{2}^{127} +$$$$16\!\cdots\!25$$$$T_{2}^{126} -$$$$27\!\cdots\!50$$$$T_{2}^{125} +$$$$49\!\cdots\!69$$$$T_{2}^{124} -$$$$88\!\cdots\!50$$$$T_{2}^{123} +$$$$13\!\cdots\!49$$$$T_{2}^{122} -$$$$19\!\cdots\!00$$$$T_{2}^{121} +$$$$31\!\cdots\!88$$$$T_{2}^{120} -$$$$54\!\cdots\!00$$$$T_{2}^{119} +$$$$87\!\cdots\!42$$$$T_{2}^{118} -$$$$13\!\cdots\!40$$$$T_{2}^{117} +$$$$19\!\cdots\!70$$$$T_{2}^{116} -$$$$28\!\cdots\!50$$$$T_{2}^{115} +$$$$46\!\cdots\!97$$$$T_{2}^{114} -$$$$74\!\cdots\!30$$$$T_{2}^{113} +$$$$96\!\cdots\!24$$$$T_{2}^{112} -$$$$11\!\cdots\!10$$$$T_{2}^{111} +$$$$17\!\cdots\!47$$$$T_{2}^{110} -$$$$29\!\cdots\!00$$$$T_{2}^{109} +$$$$36\!\cdots\!13$$$$T_{2}^{108} -$$$$42\!\cdots\!10$$$$T_{2}^{107} +$$$$60\!\cdots\!09$$$$T_{2}^{106} -$$$$80\!\cdots\!80$$$$T_{2}^{105} +$$$$98\!\cdots\!19$$$$T_{2}^{104} -$$$$14\!\cdots\!00$$$$T_{2}^{103} +$$$$15\!\cdots\!00$$$$T_{2}^{102} -$$$$86\!\cdots\!70$$$$T_{2}^{101} +$$$$15\!\cdots\!04$$$$T_{2}^{100} -$$$$33\!\cdots\!60$$$$T_{2}^{99} +$$$$38\!\cdots\!51$$$$T_{2}^{98} +$$$$48\!\cdots\!80$$$$T_{2}^{97} +$$$$73\!\cdots\!09$$$$T_{2}^{96} -$$$$52\!\cdots\!00$$$$T_{2}^{95} -$$$$10\!\cdots\!64$$$$T_{2}^{94} +$$$$21\!\cdots\!40$$$$T_{2}^{93} -$$$$12\!\cdots\!83$$$$T_{2}^{92} -$$$$68\!\cdots\!10$$$$T_{2}^{91} -$$$$24\!\cdots\!61$$$$T_{2}^{90} +$$$$30\!\cdots\!10$$$$T_{2}^{89} +$$$$67\!\cdots\!81$$$$T_{2}^{88} -$$$$90\!\cdots\!20$$$$T_{2}^{87} -$$$$14\!\cdots\!49$$$$T_{2}^{86} +$$$$38\!\cdots\!10$$$$T_{2}^{85} +$$$$22\!\cdots\!40$$$$T_{2}^{84} -$$$$24\!\cdots\!10$$$$T_{2}^{83} +$$$$22\!\cdots\!27$$$$T_{2}^{82} -$$$$30\!\cdots\!00$$$$T_{2}^{81} +$$$$78\!\cdots\!01$$$$T_{2}^{80} +$$$$89\!\cdots\!30$$$$T_{2}^{79} +$$$$56\!\cdots\!16$$$$T_{2}^{78} -$$$$99\!\cdots\!90$$$$T_{2}^{77} -$$$$52\!\cdots\!41$$$$T_{2}^{76} +$$$$11\!\cdots\!10$$$$T_{2}^{75} +$$$$51\!\cdots\!53$$$$T_{2}^{74} -$$$$10\!\cdots\!60$$$$T_{2}^{73} -$$$$71\!\cdots\!08$$$$T_{2}^{72} +$$$$13\!\cdots\!60$$$$T_{2}^{71} -$$$$66\!\cdots\!52$$$$T_{2}^{70} -$$$$28\!\cdots\!60$$$$T_{2}^{69} +$$$$34\!\cdots\!19$$$$T_{2}^{68} -$$$$92\!\cdots\!80$$$$T_{2}^{67} -$$$$57\!\cdots\!67$$$$T_{2}^{66} +$$$$16\!\cdots\!40$$$$T_{2}^{65} +$$$$10\!\cdots\!96$$$$T_{2}^{64} -$$$$20\!\cdots\!60$$$$T_{2}^{63} -$$$$91\!\cdots\!02$$$$T_{2}^{62} +$$$$17\!\cdots\!40$$$$T_{2}^{61} +$$$$56\!\cdots\!73$$$$T_{2}^{60} -$$$$18\!\cdots\!20$$$$T_{2}^{59} -$$$$62\!\cdots\!42$$$$T_{2}^{58} +$$$$64\!\cdots\!70$$$$T_{2}^{57} -$$$$49\!\cdots\!45$$$$T_{2}^{56} -$$$$14\!\cdots\!00$$$$T_{2}^{55} -$$$$58\!\cdots\!51$$$$T_{2}^{54} +$$$$43\!\cdots\!50$$$$T_{2}^{53} +$$$$41\!\cdots\!19$$$$T_{2}^{52} -$$$$14\!\cdots\!30$$$$T_{2}^{51} -$$$$20\!\cdots\!34$$$$T_{2}^{50} +$$$$17\!\cdots\!40$$$$T_{2}^{49} +$$$$53\!\cdots\!61$$$$T_{2}^{48} +$$$$58\!\cdots\!40$$$$T_{2}^{47} +$$$$44\!\cdots\!52$$$$T_{2}^{46} +$$$$33\!\cdots\!60$$$$T_{2}^{45} +$$$$32\!\cdots\!87$$$$T_{2}^{44} +$$$$34\!\cdots\!00$$$$T_{2}^{43} +$$$$32\!\cdots\!70$$$$T_{2}^{42} +$$$$28\!\cdots\!20$$$$T_{2}^{41} +$$$$26\!\cdots\!15$$$$T_{2}^{40} +$$$$25\!\cdots\!00$$$$T_{2}^{39} +$$$$24\!\cdots\!96$$$$T_{2}^{38} +$$$$22\!\cdots\!00$$$$T_{2}^{37} +$$$$18\!\cdots\!48$$$$T_{2}^{36} +$$$$14\!\cdots\!10$$$$T_{2}^{35} +$$$$95\!\cdots\!65$$$$T_{2}^{34} +$$$$60\!\cdots\!00$$$$T_{2}^{33} +$$$$35\!\cdots\!07$$$$T_{2}^{32} +$$$$18\!\cdots\!30$$$$T_{2}^{31} +$$$$92\!\cdots\!01$$$$T_{2}^{30} +$$$$42\!\cdots\!40$$$$T_{2}^{29} +$$$$17\!\cdots\!09$$$$T_{2}^{28} +$$$$69\!\cdots\!70$$$$T_{2}^{27} +$$$$25\!\cdots\!01$$$$T_{2}^{26} +$$$$85\!\cdots\!50$$$$T_{2}^{25} +$$$$26\!\cdots\!99$$$$T_{2}^{24} +$$$$75\!\cdots\!20$$$$T_{2}^{23} +$$$$19\!\cdots\!93$$$$T_{2}^{22} +$$$$44\!\cdots\!60$$$$T_{2}^{21} +$$$$88\!\cdots\!98$$$$T_{2}^{20} +$$$$16\!\cdots\!20$$$$T_{2}^{19} +$$$$26\!\cdots\!39$$$$T_{2}^{18} +$$$$42\!\cdots\!30$$$$T_{2}^{17} +$$$$62\!\cdots\!96$$$$T_{2}^{16} +$$$$83\!\cdots\!60$$$$T_{2}^{15} +$$$$98\!\cdots\!42$$$$T_{2}^{14} +$$$$10\!\cdots\!10$$$$T_{2}^{13} +$$$$12\!\cdots\!26$$$$T_{2}^{12} +$$$$12\!\cdots\!30$$$$T_{2}^{11} +$$$$11\!\cdots\!62$$$$T_{2}^{10} +$$$$10\!\cdots\!00$$$$T_{2}^{9} +$$$$96\!\cdots\!07$$$$T_{2}^{8} +$$$$73\!\cdots\!30$$$$T_{2}^{7} +$$$$53\!\cdots\!52$$$$T_{2}^{6} +$$$$35\!\cdots\!20$$$$T_{2}^{5} +$$$$19\!\cdots\!29$$$$T_{2}^{4} +$$$$10\!\cdots\!20$$$$T_{2}^{3} +$$$$49\!\cdots\!31$$$$T_{2}^{2} +$$$$14\!\cdots\!50$$$$T_{2} +$$$$19\!\cdots\!01$$">$$T_{2}^{224} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.