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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(429, [\chi])\).