Properties

 Label 429.2.bs.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$224q - 28q^{3} - 6q^{5} + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$224q - 28q^{3} - 6q^{5} + 28q^{9} - 16q^{11} - 10q^{13} - 36q^{14} + 8q^{15} - 24q^{16} + 50q^{20} + 10q^{22} - 48q^{23} - 30q^{24} + 10q^{26} + 56q^{27} + 20q^{29} + 12q^{31} - 32q^{33} - 104q^{34} + 20q^{35} + 12q^{37} + 20q^{39} - 80q^{40} + 190q^{41} + 32q^{42} - 24q^{44} + 4q^{45} - 40q^{46} - 10q^{47} + 44q^{48} - 24q^{49} - 180q^{50} - 110q^{52} + 28q^{53} - 62q^{55} - 72q^{56} + 20q^{58} - 118q^{59} + 88q^{60} - 20q^{61} + 120q^{62} - 20q^{66} - 156q^{67} - 60q^{68} + 12q^{69} - 20q^{70} + 64q^{71} - 30q^{72} + 10q^{73} - 60q^{74} - 12q^{75} + 84q^{78} + 200q^{79} + 188q^{80} + 28q^{81} - 138q^{82} - 80q^{83} - 60q^{84} + 40q^{85} - 42q^{86} - 18q^{88} + 216q^{89} - 88q^{91} + 16q^{92} + 12q^{93} - 40q^{94} - 300q^{95} - 100q^{96} + 18q^{97} - 6q^{99} + O(q^{100})$$

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 −0.954534 + 2.48665i −0.669131 0.743145i −3.78598 3.40891i −2.07220 0.328204i 2.48665 0.954534i 3.34409 0.175256i 7.34411 3.74201i −0.104528 + 0.994522i 2.79411 4.83954i
7.2 −0.834037 + 2.17274i −0.669131 0.743145i −2.53889 2.28603i 0.383236 + 0.0606986i 2.17274 0.834037i −0.733573 + 0.0384449i 2.93716 1.49656i −0.104528 + 0.994522i −0.451515 + 0.782048i
7.3 −0.578603 + 1.50731i −0.669131 0.743145i −0.450920 0.406010i −2.06394 0.326895i 1.50731 0.578603i 2.81384 0.147467i −2.00426 + 1.02122i −0.104528 + 0.994522i 1.68693 2.92185i
7.4 −0.566051 + 1.47461i −0.669131 0.743145i −0.367780 0.331151i −0.523711 0.0829477i 1.47461 0.566051i −2.90169 + 0.152071i −2.11823 + 1.07929i −0.104528 + 0.994522i 0.418763 0.725318i
7.5 −0.501760 + 1.30713i −0.669131 0.743145i 0.0294667 + 0.0265320i 2.71725 + 0.430370i 1.30713 0.501760i −2.18905 + 0.114723i −2.54451 + 1.29649i −0.104528 + 0.994522i −1.92595 + 3.33585i
7.6 −0.185537 + 0.483339i −0.669131 0.743145i 1.28710 + 1.15891i 3.92281 + 0.621313i 0.483339 0.185537i 1.27804 0.0669792i −1.72154 + 0.877171i −0.104528 + 0.994522i −1.02813 + 1.78077i
7.7 −0.0378782 + 0.0986760i −0.669131 0.743145i 1.47799 + 1.33079i −4.11879 0.652352i 0.0986760 0.0378782i 2.16416 0.113419i −0.375652 + 0.191404i −0.104528 + 0.994522i 0.220383 0.381715i
7.8 0.0793711 0.206769i −0.669131 0.743145i 1.44984 + 1.30544i 0.996265 + 0.157793i −0.206769 + 0.0793711i 3.42689 0.179596i 0.779678 0.397266i −0.104528 + 0.994522i 0.111701 0.193472i
7.9 0.0815196 0.212366i −0.669131 0.743145i 1.44784 + 1.30364i −1.64085 0.259884i −0.212366 + 0.0815196i −3.90019 + 0.204401i 0.800237 0.407741i −0.104528 + 0.994522i −0.188952 + 0.327274i
7.10 0.459872 1.19801i −0.669131 0.743145i 0.262550 + 0.236401i −2.03444 0.322223i −1.19801 + 0.459872i −3.05544 + 0.160129i 2.69070 1.37098i −0.104528 + 0.994522i −1.32161 + 2.28909i
7.11 0.463345 1.20706i −0.669131 0.743145i 0.243996 + 0.219695i 0.666783 + 0.105608i −1.20706 + 0.463345i 0.661445 0.0346649i 2.68226 1.36668i −0.104528 + 0.994522i 0.436426 0.755911i
7.12 0.701743 1.82810i −0.669131 0.743145i −1.36323 1.22746i 2.43001 + 0.384875i −1.82810 + 0.701743i 1.54304 0.0808673i 0.288919 0.147212i −0.104528 + 0.994522i 2.40883 4.17222i
7.13 0.915452 2.38484i −0.669131 0.743145i −3.36310 3.02815i 1.16930 + 0.185198i −2.38484 + 0.915452i −4.20987 + 0.220630i −5.74822 + 2.92887i −0.104528 + 0.994522i 1.51210 2.61904i
7.14 0.957096 2.49332i −0.669131 0.743145i −3.81432 3.43443i −2.91905 0.462332i −2.49332 + 0.957096i 1.75832 0.0921498i −7.45456 + 3.79829i −0.104528 + 0.994522i −3.94655 + 6.83563i
19.1 −1.55941 + 1.92572i 0.978148 + 0.207912i −0.860786 4.04968i −3.59392 + 0.569220i −1.92572 + 1.55941i 0.829321 1.27704i 4.72514 + 2.40758i 0.913545 + 0.406737i 4.50825 7.80851i
19.2 −1.39677 + 1.72487i 0.978148 + 0.207912i −0.608384 2.86222i 0.162782 0.0257822i −1.72487 + 1.39677i −2.66615 + 4.10551i 1.83157 + 0.933234i 0.913545 + 0.406737i −0.182899 + 0.316791i
19.3 −1.22830 + 1.51683i 0.978148 + 0.207912i −0.376213 1.76994i 3.41571 0.540995i −1.51683 + 1.22830i 0.116665 0.179648i −0.331315 0.168813i 0.913545 + 0.406737i −3.37493 + 5.84554i
19.4 −1.13436 + 1.40082i 0.978148 + 0.207912i −0.259701 1.22179i 0.351689 0.0557021i −1.40082 + 1.13436i 2.23633 3.44365i −1.20601 0.614491i 0.913545 + 0.406737i −0.320915 + 0.555841i
19.5 −0.622153 + 0.768295i 0.978148 + 0.207912i 0.212621 + 1.00030i −1.97898 + 0.313439i −0.768295 + 0.622153i −2.00584 + 3.08873i −2.66253 1.35662i 0.913545 + 0.406737i 0.990413 1.71545i
19.6 −0.528774 + 0.652981i 0.978148 + 0.207912i 0.269041 + 1.26574i −1.99296 + 0.315653i −0.652981 + 0.528774i 0.784828 1.20853i −2.46606 1.25652i 0.913545 + 0.406737i 0.847708 1.46827i
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.a 224
11.d odd 10 1 inner 429.2.bs.a 224
13.f odd 12 1 inner 429.2.bs.a 224
143.w even 60 1 inner 429.2.bs.a 224

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$19\!\cdots\!59$$$$T_{2}^{196} -$$$$10\!\cdots\!40$$$$T_{2}^{195} +$$$$49\!\cdots\!44$$$$T_{2}^{194} -$$$$23\!\cdots\!80$$$$T_{2}^{193} +$$$$12\!\cdots\!57$$$$T_{2}^{192} -$$$$16\!\cdots\!60$$$$T_{2}^{191} +$$$$11\!\cdots\!54$$$$T_{2}^{190} -$$$$12\!\cdots\!20$$$$T_{2}^{189} +$$$$30\!\cdots\!66$$$$T_{2}^{188} -$$$$60\!\cdots\!10$$$$T_{2}^{187} -$$$$10\!\cdots\!14$$$$T_{2}^{186} -$$$$26\!\cdots\!10$$$$T_{2}^{185} -$$$$34\!\cdots\!24$$$$T_{2}^{184} -$$$$99\!\cdots\!40$$$$T_{2}^{183} -$$$$13\!\cdots\!52$$$$T_{2}^{182} -$$$$19\!\cdots\!80$$$$T_{2}^{181} -$$$$45\!\cdots\!18$$$$T_{2}^{180} +$$$$19\!\cdots\!00$$$$T_{2}^{179} +$$$$39\!\cdots\!42$$$$T_{2}^{178} +$$$$19\!\cdots\!80$$$$T_{2}^{177} -$$$$13\!\cdots\!82$$$$T_{2}^{176} +$$$$13\!\cdots\!00$$$$T_{2}^{175} +$$$$35\!\cdots\!36$$$$T_{2}^{174} +$$$$14\!\cdots\!80$$$$T_{2}^{173} -$$$$83\!\cdots\!07$$$$T_{2}^{172} +$$$$33\!\cdots\!20$$$$T_{2}^{171} +$$$$48\!\cdots\!41$$$$T_{2}^{170} +$$$$41\!\cdots\!20$$$$T_{2}^{169} +$$$$92\!\cdots\!70$$$$T_{2}^{168} +$$$$35\!\cdots\!30$$$$T_{2}^{167} -$$$$31\!\cdots\!31$$$$T_{2}^{166} +$$$$16\!\cdots\!50$$$$T_{2}^{165} +$$$$39\!\cdots\!98$$$$T_{2}^{164} +$$$$11\!\cdots\!30$$$$T_{2}^{163} -$$$$11\!\cdots\!06$$$$T_{2}^{162} -$$$$30\!\cdots\!60$$$$T_{2}^{161} +$$$$46\!\cdots\!77$$$$T_{2}^{160} +$$$$16\!\cdots\!50$$$$T_{2}^{159} -$$$$75\!\cdots\!45$$$$T_{2}^{158} -$$$$78\!\cdots\!40$$$$T_{2}^{157} -$$$$11\!\cdots\!64$$$$T_{2}^{156} -$$$$20\!\cdots\!40$$$$T_{2}^{155} +$$$$36\!\cdots\!93$$$$T_{2}^{154} -$$$$47\!\cdots\!90$$$$T_{2}^{153} -$$$$46\!\cdots\!11$$$$T_{2}^{152} -$$$$75\!\cdots\!70$$$$T_{2}^{151} +$$$$91\!\cdots\!68$$$$T_{2}^{150} +$$$$12\!\cdots\!50$$$$T_{2}^{149} -$$$$48\!\cdots\!47$$$$T_{2}^{148} -$$$$70\!\cdots\!30$$$$T_{2}^{147} +$$$$14\!\cdots\!78$$$$T_{2}^{146} +$$$$55\!\cdots\!00$$$$T_{2}^{145} +$$$$67\!\cdots\!36$$$$T_{2}^{144} +$$$$11\!\cdots\!60$$$$T_{2}^{143} +$$$$19\!\cdots\!24$$$$T_{2}^{142} +$$$$11\!\cdots\!50$$$$T_{2}^{141} +$$$$26\!\cdots\!73$$$$T_{2}^{140} +$$$$34\!\cdots\!90$$$$T_{2}^{139} +$$$$14\!\cdots\!70$$$$T_{2}^{138} +$$$$12\!\cdots\!30$$$$T_{2}^{137} +$$$$42\!\cdots\!90$$$$T_{2}^{136} +$$$$48\!\cdots\!40$$$$T_{2}^{135} -$$$$39\!\cdots\!27$$$$T_{2}^{134} +$$$$17\!\cdots\!10$$$$T_{2}^{133} +$$$$28\!\cdots\!01$$$$T_{2}^{132} +$$$$38\!\cdots\!40$$$$T_{2}^{131} -$$$$33\!\cdots\!62$$$$T_{2}^{130} -$$$$20\!\cdots\!10$$$$T_{2}^{129} -$$$$27\!\cdots\!73$$$$T_{2}^{128} -$$$$35\!\cdots\!70$$$$T_{2}^{127} -$$$$11\!\cdots\!15$$$$T_{2}^{126} -$$$$44\!\cdots\!30$$$$T_{2}^{125} -$$$$10\!\cdots\!15$$$$T_{2}^{124} -$$$$16\!\cdots\!90$$$$T_{2}^{123} -$$$$21\!\cdots\!35$$$$T_{2}^{122} -$$$$44\!\cdots\!20$$$$T_{2}^{121} -$$$$10\!\cdots\!56$$$$T_{2}^{120} -$$$$18\!\cdots\!60$$$$T_{2}^{119} -$$$$21\!\cdots\!94$$$$T_{2}^{118} -$$$$19\!\cdots\!00$$$$T_{2}^{117} -$$$$25\!\cdots\!02$$$$T_{2}^{116} -$$$$30\!\cdots\!30$$$$T_{2}^{115} +$$$$29\!\cdots\!21$$$$T_{2}^{114} +$$$$28\!\cdots\!90$$$$T_{2}^{113} +$$$$79\!\cdots\!16$$$$T_{2}^{112} +$$$$16\!\cdots\!30$$$$T_{2}^{111} +$$$$31\!\cdots\!59$$$$T_{2}^{110} +$$$$65\!\cdots\!00$$$$T_{2}^{109} +$$$$13\!\cdots\!49$$$$T_{2}^{108} +$$$$27\!\cdots\!10$$$$T_{2}^{107} +$$$$51\!\cdots\!29$$$$T_{2}^{106} +$$$$93\!\cdots\!60$$$$T_{2}^{105} +$$$$17\!\cdots\!39$$$$T_{2}^{104} +$$$$30\!\cdots\!60$$$$T_{2}^{103} +$$$$51\!\cdots\!88$$$$T_{2}^{102} +$$$$81\!\cdots\!50$$$$T_{2}^{101} +$$$$12\!\cdots\!84$$$$T_{2}^{100} +$$$$18\!\cdots\!60$$$$T_{2}^{99} +$$$$25\!\cdots\!91$$$$T_{2}^{98} +$$$$33\!\cdots\!20$$$$T_{2}^{97} +$$$$41\!\cdots\!73$$$$T_{2}^{96} +$$$$49\!\cdots\!20$$$$T_{2}^{95} +$$$$54\!\cdots\!68$$$$T_{2}^{94} +$$$$53\!\cdots\!40$$$$T_{2}^{93} +$$$$42\!\cdots\!65$$$$T_{2}^{92} +$$$$20\!\cdots\!10$$$$T_{2}^{91} -$$$$13\!\cdots\!73$$$$T_{2}^{90} -$$$$60\!\cdots\!50$$$$T_{2}^{89} -$$$$13\!\cdots\!39$$$$T_{2}^{88} -$$$$20\!\cdots\!00$$$$T_{2}^{87} -$$$$27\!\cdots\!25$$$$T_{2}^{86} -$$$$32\!\cdots\!10$$$$T_{2}^{85} -$$$$40\!\cdots\!76$$$$T_{2}^{84} -$$$$45\!\cdots\!70$$$$T_{2}^{83} -$$$$46\!\cdots\!69$$$$T_{2}^{82} -$$$$40\!\cdots\!80$$$$T_{2}^{81} -$$$$30\!\cdots\!87$$$$T_{2}^{80} -$$$$16\!\cdots\!50$$$$T_{2}^{79} +$$$$66\!\cdots\!16$$$$T_{2}^{78} +$$$$17\!\cdots\!10$$$$T_{2}^{77} +$$$$35\!\cdots\!23$$$$T_{2}^{76} +$$$$44\!\cdots\!10$$$$T_{2}^{75} +$$$$73\!\cdots\!81$$$$T_{2}^{74} +$$$$81\!\cdots\!60$$$$T_{2}^{73} +$$$$12\!\cdots\!72$$$$T_{2}^{72} +$$$$13\!\cdots\!20$$$$T_{2}^{71} +$$$$12\!\cdots\!60$$$$T_{2}^{70} +$$$$76\!\cdots\!00$$$$T_{2}^{69} +$$$$80\!\cdots\!95$$$$T_{2}^{68} +$$$$11\!\cdots\!60$$$$T_{2}^{67} +$$$$16\!\cdots\!69$$$$T_{2}^{66} +$$$$16\!\cdots\!60$$$$T_{2}^{65} +$$$$15\!\cdots\!88$$$$T_{2}^{64} +$$$$11\!\cdots\!80$$$$T_{2}^{63} +$$$$10\!\cdots\!22$$$$T_{2}^{62} +$$$$11\!\cdots\!80$$$$T_{2}^{61} +$$$$11\!\cdots\!65$$$$T_{2}^{60} +$$$$51\!\cdots\!20$$$$T_{2}^{59} +$$$$12\!\cdots\!02$$$$T_{2}^{58} +$$$$14\!\cdots\!10$$$$T_{2}^{57} +$$$$43\!\cdots\!31$$$$T_{2}^{56} +$$$$41\!\cdots\!80$$$$T_{2}^{55} +$$$$22\!\cdots\!05$$$$T_{2}^{54} +$$$$14\!\cdots\!70$$$$T_{2}^{53} +$$$$13\!\cdots\!47$$$$T_{2}^{52} +$$$$96\!\cdots\!10$$$$T_{2}^{51} +$$$$47\!\cdots\!14$$$$T_{2}^{50} +$$$$13\!\cdots\!20$$$$T_{2}^{49} -$$$$16\!\cdots\!31$$$$T_{2}^{48} -$$$$77\!\cdots\!00$$$$T_{2}^{47} +$$$$12\!\cdots\!96$$$$T_{2}^{46} -$$$$76\!\cdots\!80$$$$T_{2}^{45} -$$$$20\!\cdots\!49$$$$T_{2}^{44} -$$$$11\!\cdots\!20$$$$T_{2}^{43} -$$$$21\!\cdots\!62$$$$T_{2}^{42} -$$$$64\!\cdots\!40$$$$T_{2}^{41} -$$$$12\!\cdots\!61$$$$T_{2}^{40} -$$$$87\!\cdots\!00$$$$T_{2}^{39} -$$$$20\!\cdots\!24$$$$T_{2}^{38} +$$$$34\!\cdots\!00$$$$T_{2}^{37} +$$$$19\!\cdots\!88$$$$T_{2}^{36} +$$$$68\!\cdots\!70$$$$T_{2}^{35} +$$$$12\!\cdots\!17$$$$T_{2}^{34} +$$$$10\!\cdots\!60$$$$T_{2}^{33} +$$$$51\!\cdots\!39$$$$T_{2}^{32} +$$$$22\!\cdots\!50$$$$T_{2}^{31} +$$$$91\!\cdots\!93$$$$T_{2}^{30} +$$$$32\!\cdots\!40$$$$T_{2}^{29} +$$$$13\!\cdots\!21$$$$T_{2}^{28} +$$$$51\!\cdots\!90$$$$T_{2}^{27} +$$$$13\!\cdots\!41$$$$T_{2}^{26} +$$$$40\!\cdots\!50$$$$T_{2}^{25} +$$$$18\!\cdots\!71$$$$T_{2}^{24} +$$$$75\!\cdots\!00$$$$T_{2}^{23} +$$$$24\!\cdots\!45$$$$T_{2}^{22} +$$$$87\!\cdots\!20$$$$T_{2}^{21} +$$$$32\!\cdots\!06$$$$T_{2}^{20} +$$$$99\!\cdots\!60$$$$T_{2}^{19} +$$$$29\!\cdots\!43$$$$T_{2}^{18} +$$$$90\!\cdots\!10$$$$T_{2}^{17} +$$$$23\!\cdots\!88$$$$T_{2}^{16} +$$$$58\!\cdots\!60$$$$T_{2}^{15} +$$$$16\!\cdots\!66$$$$T_{2}^{14} +$$$$38\!\cdots\!10$$$$T_{2}^{13} +$$$$79\!\cdots\!86$$$$T_{2}^{12} +$$$$19\!\cdots\!90$$$$T_{2}^{11} +$$$$42\!\cdots\!54$$$$T_{2}^{10} +$$$$74\!\cdots\!60$$$$T_{2}^{9} +$$$$14\!\cdots\!71$$$$T_{2}^{8} +$$$$30\!\cdots\!70$$$$T_{2}^{7} +$$$$47\!\cdots\!36$$$$T_{2}^{6} +$$$$72\!\cdots\!40$$$$T_{2}^{5} +$$$$12\!\cdots\!61$$$$T_{2}^{4} +$$$$17\!\cdots\!00$$$$T_{2}^{3} +$$$$18\!\cdots\!27$$$$T_{2}^{2} +$$$$12\!\cdots\!70$$$$T_{2} +$$$$46\!\cdots\!21$$">$$T_{2}^{224} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.