Newform orbit 429.2.bs.b

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 224q + 28q^{3} + 6q^{5} + 28q^{9} + O(q^{10}) \)

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

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