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 $4$

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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.

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

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:

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