Newform orbit 429.2.bn.b

• Label: 429.2.bn.b
• Level: $429$
• Weight: $2$
• Character orbit: 429.bn
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112q + 14q^{3} - 8q^{4} + 14q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1.08659	−	2.44051i	0.669131	+	0.743145i	−3.43718	+	3.81737i	−1.96001	−	2.69773i	1.08659	−	2.44051i	−2.63121	−	2.36915i	7.96970	+	2.58951i	−0.104528	+	0.994522i	−4.45412	+	7.71476i
4.2	−0.976080	−	2.19231i	0.669131	+	0.743145i	−2.51524	+	2.79345i	0.603494	+	0.830639i	0.976080	−	2.19231i	0.422137	+	0.380094i	4.01454	+	1.30440i	−0.104528	+	0.994522i	1.23196	−	2.13382i
4.3	−0.819494	−	1.84061i	0.669131	+	0.743145i	−1.37803	+	1.53046i	−1.15996	−	1.59655i	0.819494	−	1.84061i	3.61282	+	3.25299i	0.113886	+	0.0370037i	−0.104528	+	0.994522i	−1.98805	+	3.44340i
4.4	−0.727584	−	1.63418i	0.669131	+	0.743145i	−0.802903	+	0.891714i	−0.105484	−	0.145187i	0.727584	−	1.63418i	−2.03695	−	1.83408i	−1.36116	−	0.442268i	−0.104528	+	0.994522i	−0.160512	+	0.278015i
4.5	−0.425286	−	0.955208i	0.669131	+	0.743145i	0.606707	−	0.673817i	2.35366	+	3.23953i	0.425286	−	0.955208i	−1.51333	−	1.36261i	−2.89052	−	0.939186i	−0.104528	+	0.994522i	2.09345	−	3.62596i
4.6	−0.288687	−	0.648403i	0.669131	+	0.743145i	1.00118	−	1.11192i	−2.40424	−	3.30915i	0.288687	−	0.648403i	0.504508	+	0.454261i	−2.36005	−	0.766827i	−0.104528	+	0.994522i	−1.45159	+	2.51422i
4.7	0.0203023	+	0.0455997i	0.669131	+	0.743145i	1.33659	−	1.48444i	1.77101	+	2.43759i	−0.0203023	+	0.0455997i	0.890300	+	0.801630i	0.189770	+	0.0616600i	−0.104528	+	0.994522i	−0.0751978	+	0.130246i
4.8	0.0299019	+	0.0671608i	0.669131	+	0.743145i	1.33464	−	1.48227i	0.0495558	+	0.0682077i	−0.0299019	+	0.0671608i	1.42900	+	1.28668i	0.279296	+	0.0907488i	−0.104528	+	0.994522i	−0.00309907	+	0.00536775i
4.9	0.293907	+	0.660125i	0.669131	+	0.743145i	0.988877	−	1.09826i	−1.15395	−	1.58828i	−0.293907	+	0.660125i	−3.62548	−	3.26440i	2.39009	+	0.776586i	−0.104528	+	0.994522i	0.709310	−	1.22856i
4.10	0.434661	+	0.976265i	0.669131	+	0.743145i	0.574098	−	0.637600i	0.194734	+	0.268028i	−0.434661	+	0.976265i	1.78328	+	1.60567i	2.90471	+	0.943797i	−0.104528	+	0.994522i	−0.177023	+	0.306614i
4.11	0.644170	+	1.44683i	0.669131	+	0.743145i	−0.340099	+	0.377718i	−1.31195	−	1.80574i	−0.644170	+	1.44683i	1.36339	+	1.22760i	2.24690	+	0.730062i	−0.104528	+	0.994522i	1.76748	−	3.06137i
4.12	0.772619	+	1.73533i	0.669131	+	0.743145i	−1.07617	+	1.19521i	1.38712	+	1.90920i	−0.772619	+	1.73533i	−2.05909	−	1.85401i	0.707612	+	0.229917i	−0.104528	+	0.994522i	−2.24139	+	3.88220i
4.13	1.04016	+	2.33623i	0.669131	+	0.743145i	−3.03778	+	3.37380i	−0.492512	−	0.677885i	−1.04016	+	2.33623i	3.64547	+	3.28240i	−6.17741	−	2.00716i	−0.104528	+	0.994522i	1.07140	−	1.85573i
4.14	1.08800	+	2.44369i	0.669131	+	0.743145i	−3.44962	+	3.83119i	0.581262	+	0.800038i	−1.08800	+	2.44369i	−1.78483	−	1.60707i	−8.02736	−	2.60825i	−0.104528	+	0.994522i	−1.32263	+	2.29087i
49.1	−1.91017	−	1.71993i	−0.104528	+	0.994522i	0.481555	+	4.58169i	3.19249	+	1.03730i	1.91017	−	1.71993i	−2.98254	+	0.313478i	3.93864	−	5.42108i	−0.978148	−	0.207912i	−4.31412	−	7.47228i
49.2	−1.64800	−	1.48387i	−0.104528	+	0.994522i	0.304992	+	2.90181i	−1.57487	−	0.511707i	1.64800	−	1.48387i	1.08496	−	0.114034i	1.19632	−	1.64659i	−0.978148	−	0.207912i	1.83609	+	3.18020i
49.3	−1.60557	−	1.44566i	−0.104528	+	0.994522i	0.278861	+	2.65319i	−0.655963	−	0.213135i	1.60557	−	1.44566i	0.714969	−	0.0751462i	0.848055	−	1.16725i	−0.978148	−	0.207912i	0.745073	+	1.29051i
49.4	−0.977278	−	0.879945i	−0.104528	+	0.994522i	−0.0282879	−	0.269141i	1.70562	+	0.554188i	0.977278	−	0.879945i	2.04084	−	0.214501i	−1.75513	+	2.41573i	−0.978148	−	0.207912i	−1.17921	−	2.04245i
49.5	−0.646484	−	0.582097i	−0.104528	+	0.994522i	−0.129952	−	1.23641i	−2.84399	−	0.924069i	0.646484	−	0.582097i	0.352201	−	0.0370178i	−1.65836	+	2.28254i	−0.978148	−	0.207912i	1.30070	+	2.25288i
49.6	−0.429748	−	0.386946i	−0.104528	+	0.994522i	−0.174102	−	1.65647i	0.866536	+	0.281555i	0.429748	−	0.386946i	−3.42237	+	0.359706i	−1.24596	+	1.71491i	−0.978148	−	0.207912i	−0.263445	−	0.456300i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
13.e	even	6	1	inner
143.u	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.bn.b	✓	112
11.c	even	5	1	inner	429.2.bn.b	✓	112
13.e	even	6	1	inner	429.2.bn.b	✓	112
143.u	even	30	1	inner	429.2.bn.b	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.bn.b	✓	112	1.a	even	1	1	trivial
429.2.bn.b	✓	112	11.c	even	5	1	inner
429.2.bn.b	✓	112	13.e	even	6	1	inner
429.2.bn.b	✓	112	143.u	even	30	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(96\!\cdots\!80\)\( T_{2}^{82} + \)\(46\!\cdots\!16\)\( T_{2}^{81} + \)\(44\!\cdots\!17\)\( T_{2}^{80} + \)\(94\!\cdots\!05\)\( T_{2}^{79} + \)\(48\!\cdots\!42\)\( T_{2}^{78} - \)\(39\!\cdots\!65\)\( T_{2}^{77} - \)\(56\!\cdots\!63\)\( T_{2}^{76} - \)\(22\!\cdots\!10\)\( T_{2}^{75} - \)\(35\!\cdots\!71\)\( T_{2}^{74} + \)\(36\!\cdots\!44\)\( T_{2}^{73} - \)\(85\!\cdots\!30\)\( T_{2}^{72} + \)\(45\!\cdots\!56\)\( T_{2}^{71} + \)\(48\!\cdots\!45\)\( T_{2}^{70} + \)\(22\!\cdots\!20\)\( T_{2}^{69} + \)\(12\!\cdots\!24\)\( T_{2}^{68} - \)\(10\!\cdots\!99\)\( T_{2}^{67} + \)\(52\!\cdots\!69\)\( T_{2}^{66} - \)\(36\!\cdots\!10\)\( T_{2}^{65} + \)\(82\!\cdots\!85\)\( T_{2}^{64} + \)\(66\!\cdots\!27\)\( T_{2}^{63} - \)\(19\!\cdots\!49\)\( T_{2}^{62} + \)\(44\!\cdots\!77\)\( T_{2}^{61} - \)\(13\!\cdots\!54\)\( T_{2}^{60} + \)\(14\!\cdots\!12\)\( T_{2}^{59} - \)\(27\!\cdots\!34\)\( T_{2}^{58} + \)\(11\!\cdots\!79\)\( T_{2}^{57} - \)\(59\!\cdots\!42\)\( T_{2}^{56} - \)\(48\!\cdots\!61\)\( T_{2}^{55} + \)\(11\!\cdots\!41\)\( T_{2}^{54} - \)\(19\!\cdots\!23\)\( T_{2}^{53} + \)\(36\!\cdots\!41\)\( T_{2}^{52} - \)\(37\!\cdots\!26\)\( T_{2}^{51} + \)\(66\!\cdots\!76\)\( T_{2}^{50} - \)\(43\!\cdots\!31\)\( T_{2}^{49} + \)\(70\!\cdots\!05\)\( T_{2}^{48} - \)\(89\!\cdots\!42\)\( T_{2}^{47} + \)\(82\!\cdots\!71\)\( T_{2}^{46} + \)\(70\!\cdots\!75\)\( T_{2}^{45} - \)\(10\!\cdots\!31\)\( T_{2}^{44} + \)\(12\!\cdots\!15\)\( T_{2}^{43} - \)\(16\!\cdots\!70\)\( T_{2}^{42} + \)\(65\!\cdots\!37\)\( T_{2}^{41} - \)\(11\!\cdots\!31\)\( T_{2}^{40} - \)\(59\!\cdots\!00\)\( T_{2}^{39} + \)\(10\!\cdots\!17\)\( T_{2}^{38} - \)\(12\!\cdots\!11\)\( T_{2}^{37} + \)\(11\!\cdots\!72\)\( T_{2}^{36} - \)\(82\!\cdots\!20\)\( T_{2}^{35} + \)\(14\!\cdots\!67\)\( T_{2}^{34} - \)\(15\!\cdots\!91\)\( T_{2}^{33} + \)\(11\!\cdots\!26\)\( T_{2}^{32} + \)\(33\!\cdots\!72\)\( T_{2}^{31} + \)\(55\!\cdots\!59\)\( T_{2}^{30} + \)\(42\!\cdots\!60\)\( T_{2}^{29} + \)\(23\!\cdots\!99\)\( T_{2}^{28} + \)\(16\!\cdots\!57\)\( T_{2}^{27} - \)\(20\!\cdots\!21\)\( T_{2}^{26} - \)\(88\!\cdots\!56\)\( T_{2}^{25} - \)\(16\!\cdots\!01\)\( T_{2}^{24} - \)\(90\!\cdots\!16\)\( T_{2}^{23} - \)\(45\!\cdots\!71\)\( T_{2}^{22} - \)\(19\!\cdots\!74\)\( T_{2}^{21} + \)\(13\!\cdots\!05\)\( T_{2}^{20} + \)\(10\!\cdots\!50\)\( T_{2}^{19} + \)\(15\!\cdots\!46\)\( T_{2}^{18} + \)\(57\!\cdots\!09\)\( T_{2}^{17} + \)\(50\!\cdots\!19\)\( T_{2}^{16} + \)\(10\!\cdots\!45\)\( T_{2}^{15} + \)\(65\!\cdots\!51\)\( T_{2}^{14} + \)\(11\!\cdots\!35\)\( T_{2}^{13} + \)\(50\!\cdots\!45\)\( T_{2}^{12} - \)\(22\!\cdots\!27\)\( T_{2}^{11} + \)\(40\!\cdots\!36\)\( T_{2}^{10} - \)\(43\!\cdots\!37\)\( T_{2}^{9} + \)\(31\!\cdots\!22\)\( T_{2}^{8} - \)\(15\!\cdots\!30\)\( T_{2}^{7} + 50623041572 T_{2}^{6} - 1012957494 T_{2}^{5} + 8921117 T_{2}^{4} - 70728 T_{2}^{3} + 3515 T_{2}^{2} - 21 T_{2} + 1 \)">\(T_{2}^{112} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).