Newform orbit 429.2.bn.a

• Label: 429.2.bn.a
• 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.

\(\operatorname{Tr}(f)(q) = \) \( 112q - 14q^{3} - 20q^{4} + 14q^{9} + O(q^{10}) \)

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	−1.05844	−	2.37729i	−0.669131	−	0.743145i	−3.19297	+	3.54616i	0.696965	+	0.959290i	−1.05844	+	2.37729i	1.37323	+	1.23646i	6.86002	+	2.22895i	−0.104528	+	0.994522i	1.54282	−	2.67224i
4.2	−0.866606	−	1.94643i	−0.669131	−	0.743145i	−1.69932	+	1.88729i	1.80442	+	2.48357i	−0.866606	+	1.94643i	−3.61054	−	3.25094i	1.09341	+	0.355271i	−0.104528	+	0.994522i	3.27037	−	5.66445i
4.3	−0.763963	−	1.71589i	−0.669131	−	0.743145i	−1.02237	+	1.13546i	−1.07133	−	1.47456i	−0.763963	+	1.71589i	1.14179	+	1.02807i	−0.843308	−	0.274007i	−0.104528	+	0.994522i	−1.71172	+	2.96479i
4.4	−0.688193	−	1.54571i	−0.669131	−	0.743145i	−0.577339	+	0.641200i	1.57124	+	2.16263i	−0.688193	+	1.54571i	1.25124	+	1.12663i	−1.82992	−	0.594578i	−0.104528	+	0.994522i	2.26148	−	3.91699i
4.5	−0.672174	−	1.50973i	−0.669131	−	0.743145i	−0.489197	+	0.543308i	−2.52085	−	3.46966i	−0.672174	+	1.50973i	−2.16497	−	1.94935i	−1.99436	−	0.648008i	−0.104528	+	0.994522i	−3.54378	+	6.13802i
4.6	−0.177986	−	0.399764i	−0.669131	−	0.743145i	1.21013	−	1.34398i	0.0311778	+	0.0429126i	−0.177986	+	0.399764i	−0.860670	−	0.774951i	−1.58502	−	0.515004i	−0.104528	+	0.994522i	0.0116057	−	0.0201016i
4.7	−0.0803440	−	0.180456i	−0.669131	−	0.743145i	1.31215	−	1.45729i	0.296705	+	0.408379i	−0.0803440	+	0.180456i	3.31863	+	2.98811i	−0.744131	−	0.241783i	−0.104528	+	0.994522i	0.0498559	−	0.0863529i
4.8	0.250968	+	0.563684i	−0.669131	−	0.743145i	1.08351	−	1.20336i	−1.79971	−	2.47709i	0.250968	−	0.563684i	−2.23596	−	2.01327i	2.12390	+	0.690096i	−0.104528	+	0.994522i	0.944626	−	1.63614i
4.9	0.275867	+	0.619607i	−0.669131	−	0.743145i	1.03045	−	1.14443i	−1.77061	−	2.43703i	0.275867	−	0.619607i	3.22654	+	2.90519i	2.28346	+	0.741942i	−0.104528	+	0.994522i	1.02155	−	1.76938i
4.10	0.386229	+	0.867484i	−0.669131	−	0.743145i	0.734905	−	0.816195i	2.09746	+	2.88691i	0.386229	−	0.867484i	0.681566	+	0.613685i	2.79809	+	0.909153i	−0.104528	+	0.994522i	−1.69425	+	2.93453i
4.11	0.534181	+	1.19979i	−0.669131	−	0.743145i	0.184114	−	0.204479i	1.16055	+	1.59736i	0.534181	−	1.19979i	−2.62438	−	2.36301i	2.84179	+	0.923354i	−0.104528	+	0.994522i	−1.29655	+	2.24569i
4.12	0.883617	+	1.98464i	−0.669131	−	0.743145i	−1.81974	+	2.02103i	−1.20025	−	1.65200i	0.883617	−	1.98464i	−2.60068	−	2.34166i	−1.48670	−	0.483058i	−0.104528	+	0.994522i	2.21806	−	3.84180i
4.13	0.896442	+	2.01344i	−0.669131	−	0.743145i	−1.91208	+	2.12358i	0.00844073	+	0.0116177i	0.896442	−	2.01344i	1.50438	+	1.35455i	−1.79754	−	0.584056i	−0.104528	+	0.994522i	−0.0158249	+	0.0274095i
4.14	1.08040	+	2.42662i	−0.669131	−	0.743145i	−3.38297	+	3.75717i	2.34307	+	3.22496i	1.08040	−	2.42662i	1.59982	+	1.44048i	−7.71968	−	2.50828i	−0.104528	+	0.994522i	−5.29430	+	9.17000i
49.1	−1.97287	−	1.77638i	0.104528	−	0.994522i	0.527630	+	5.02007i	1.64057	+	0.533054i	−1.97287	+	1.77638i	−2.55155	+	0.268179i	4.75574	−	6.54571i	−0.978148	−	0.207912i	−2.28973	−	3.96592i
49.2	−1.80408	−	1.62440i	0.104528	−	0.994522i	0.406968	+	3.87204i	−2.99760	−	0.973979i	−1.80408	+	1.62440i	1.24847	−	0.131220i	2.70169	−	3.71856i	−0.978148	−	0.207912i	3.82577	+	6.62643i
49.3	−1.54435	−	1.39054i	0.104528	−	0.994522i	0.242362	+	2.30592i	1.99851	+	0.649356i	−1.54435	+	1.39054i	2.22364	−	0.233714i	0.389193	−	0.535678i	−0.978148	−	0.207912i	−2.18345	−	3.78184i
49.4	−1.38170	−	1.24409i	0.104528	−	0.994522i	0.152285	+	1.44889i	−1.73448	−	0.563568i	−1.38170	+	1.24409i	−4.66758	+	0.490582i	−0.593557	+	0.816960i	−0.978148	−	0.207912i	1.69541	+	2.93654i
49.5	−1.04336	−	0.939450i	0.104528	−	0.994522i	−0.00301284	−	0.0286652i	1.68655	+	0.547992i	−1.04336	+	0.939450i	−0.0876448	+	0.00921184i	−1.67427	+	2.30444i	−0.978148	−	0.207912i	−1.24487	−	2.15618i
49.6	−0.280336	−	0.252415i	0.104528	−	0.994522i	−0.194182	−	1.84752i	−2.49942	−	0.812112i	−0.280336	+	0.252415i	−1.73582	+	0.182442i	−0.855366	+	1.17731i	−0.978148	−	0.207912i	0.495688	+	0.858557i

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.a	✓	112
11.c	even	5	1	inner	429.2.bn.a	✓	112
13.e	even	6	1	inner	429.2.bn.a	✓	112
143.u	even	30	1	inner	429.2.bn.a	✓	112

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(25\!\cdots\!34\)\( T_{2}^{84} - 603643065171 T_{2}^{83} - \)\(14\!\cdots\!86\)\( T_{2}^{82} + \)\(10\!\cdots\!60\)\( T_{2}^{81} + \)\(53\!\cdots\!19\)\( T_{2}^{80} + \)\(78\!\cdots\!51\)\( T_{2}^{79} + \)\(32\!\cdots\!46\)\( T_{2}^{78} + \)\(18\!\cdots\!57\)\( T_{2}^{77} + \)\(73\!\cdots\!97\)\( T_{2}^{76} - \)\(51\!\cdots\!20\)\( T_{2}^{75} - \)\(15\!\cdots\!67\)\( T_{2}^{74} - \)\(49\!\cdots\!52\)\( T_{2}^{73} - \)\(18\!\cdots\!08\)\( T_{2}^{72} - \)\(15\!\cdots\!88\)\( T_{2}^{71} - \)\(74\!\cdots\!69\)\( T_{2}^{70} - \)\(17\!\cdots\!66\)\( T_{2}^{69} - \)\(12\!\cdots\!08\)\( T_{2}^{68} + \)\(54\!\cdots\!09\)\( T_{2}^{67} + \)\(35\!\cdots\!09\)\( T_{2}^{66} + \)\(40\!\cdots\!16\)\( T_{2}^{65} + \)\(30\!\cdots\!93\)\( T_{2}^{64} + \)\(14\!\cdots\!81\)\( T_{2}^{63} + \)\(98\!\cdots\!59\)\( T_{2}^{62} + \)\(43\!\cdots\!35\)\( T_{2}^{61} + \)\(15\!\cdots\!50\)\( T_{2}^{60} + \)\(17\!\cdots\!02\)\( T_{2}^{59} - \)\(13\!\cdots\!76\)\( T_{2}^{58} + \)\(65\!\cdots\!77\)\( T_{2}^{57} - \)\(14\!\cdots\!12\)\( T_{2}^{56} + \)\(13\!\cdots\!97\)\( T_{2}^{55} - \)\(34\!\cdots\!25\)\( T_{2}^{54} - \)\(34\!\cdots\!83\)\( T_{2}^{53} + \)\(45\!\cdots\!53\)\( T_{2}^{52} - \)\(12\!\cdots\!80\)\( T_{2}^{51} + \)\(22\!\cdots\!76\)\( T_{2}^{50} - \)\(39\!\cdots\!43\)\( T_{2}^{49} + \)\(56\!\cdots\!95\)\( T_{2}^{48} - \)\(45\!\cdots\!42\)\( T_{2}^{47} + \)\(28\!\cdots\!57\)\( T_{2}^{46} + \)\(75\!\cdots\!11\)\( T_{2}^{45} - \)\(18\!\cdots\!95\)\( T_{2}^{44} + \)\(40\!\cdots\!19\)\( T_{2}^{43} - \)\(60\!\cdots\!16\)\( T_{2}^{42} + \)\(79\!\cdots\!47\)\( T_{2}^{41} - \)\(88\!\cdots\!43\)\( T_{2}^{40} + \)\(61\!\cdots\!92\)\( T_{2}^{39} - \)\(28\!\cdots\!67\)\( T_{2}^{38} - \)\(13\!\cdots\!31\)\( T_{2}^{37} + \)\(31\!\cdots\!46\)\( T_{2}^{36} - \)\(53\!\cdots\!34\)\( T_{2}^{35} + \)\(73\!\cdots\!63\)\( T_{2}^{34} - \)\(83\!\cdots\!69\)\( T_{2}^{33} + \)\(80\!\cdots\!84\)\( T_{2}^{32} - \)\(63\!\cdots\!68\)\( T_{2}^{31} + \)\(41\!\cdots\!91\)\( T_{2}^{30} - \)\(20\!\cdots\!32\)\( T_{2}^{29} + \)\(71\!\cdots\!47\)\( T_{2}^{28} - \)\(86\!\cdots\!39\)\( T_{2}^{27} - \)\(10\!\cdots\!05\)\( T_{2}^{26} + \)\(10\!\cdots\!36\)\( T_{2}^{25} - \)\(51\!\cdots\!75\)\( T_{2}^{24} + \)\(18\!\cdots\!68\)\( T_{2}^{23} - \)\(42\!\cdots\!17\)\( T_{2}^{22} + \)\(85\!\cdots\!64\)\( T_{2}^{21} + \)\(12\!\cdots\!47\)\( T_{2}^{20} - \)\(26\!\cdots\!48\)\( T_{2}^{19} + \)\(14\!\cdots\!02\)\( T_{2}^{18} - \)\(58\!\cdots\!17\)\( T_{2}^{17} + \)\(65\!\cdots\!71\)\( T_{2}^{16} - \)\(17\!\cdots\!05\)\( T_{2}^{15} + \)\(37\!\cdots\!15\)\( T_{2}^{14} - \)\(11\!\cdots\!45\)\( T_{2}^{13} + \)\(71\!\cdots\!85\)\( T_{2}^{12} + \)\(43\!\cdots\!75\)\( T_{2}^{11} + \)\(11\!\cdots\!50\)\( T_{2}^{10} + \)\(27\!\cdots\!25\)\( T_{2}^{9} + \)\(37\!\cdots\!00\)\( T_{2}^{8} + \)\(58\!\cdots\!00\)\( T_{2}^{7} + \)\(10\!\cdots\!50\)\( T_{2}^{6} + \)\(16\!\cdots\!00\)\( T_{2}^{5} + 292490938375 T_{2}^{4} + 26808127500 T_{2}^{3} + 2455619375 T_{2}^{2} + 152233125 T_{2} + 9150625 \)">\(T_{2}^{112} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).