Newform orbit 429.2.bg.a

• Label: 429.2.bg.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.bg
• 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.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112q - 14q^{3} + 20q^{4} + 6q^{5} + 24q^{8} + 14q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−1.80037	+	1.99951i	0.104528	−	0.994522i	−0.547661	−	5.21064i	−0.759776	+	2.33835i	1.80037	+	1.99951i	0.0209204	+	0.199044i	7.05123	+	5.12302i	−0.978148	−	0.207912i	−3.30768	−	5.72907i
16.2	−1.68093	+	1.86686i	0.104528	−	0.994522i	−0.450589	−	4.28707i	1.24954	−	3.84568i	1.68093	+	1.86686i	0.0277297	+	0.263830i	4.69608	+	3.41190i	−0.978148	−	0.207912i	5.07897	+	8.79703i
16.3	−1.46128	+	1.62292i	0.104528	−	0.994522i	−0.289460	−	2.75403i	−0.299446	+	0.921600i	1.46128	+	1.62292i	−0.235463	−	2.24028i	1.35900	+	0.987368i	−0.978148	−	0.207912i	−1.05810	−	1.83269i
16.4	−0.990763	+	1.10035i	0.104528	−	0.994522i	−0.0201101	−	0.191335i	−0.983732	+	3.02762i	0.990763	+	1.10035i	0.365713	+	3.47953i	−2.16532	−	1.57319i	−0.978148	−	0.207912i	−2.35680	−	4.08210i
16.5	−0.946617	+	1.05132i	0.104528	−	0.994522i	−0.000142744	−	0.00135811i	0.776847	−	2.39089i	0.946617	+	1.05132i	−0.131132	−	1.24764i	−2.28746	−	1.66194i	−0.978148	−	0.207912i	1.77822	+	3.07997i
16.6	−0.491663	+	0.546047i	0.104528	−	0.994522i	0.152622	+	1.45210i	−0.00147074	+	0.00452647i	0.491663	+	0.546047i	−0.428655	−	4.07838i	−2.05685	−	1.49439i	−0.978148	−	0.207912i	−0.00174856	−	0.00302859i
16.7	−0.0978317	+	0.108653i	0.104528	−	0.994522i	0.206822	+	1.96778i	−0.621977	+	1.91425i	0.0978317	+	0.108653i	0.00250914	+	0.0238729i	−0.470608	−	0.341917i	−0.978148	−	0.207912i	−0.147140	−	0.254854i
16.8	0.244102	−	0.271103i	0.104528	−	0.994522i	0.195146	+	1.85669i	0.211790	−	0.651823i	−0.244102	−	0.271103i	0.465929	+	4.43302i	1.14126	+	0.829171i	−0.978148	−	0.207912i	−0.125013	−	0.216528i
16.9	0.502391	−	0.557961i	0.104528	−	0.994522i	0.150132	+	1.42841i	−0.0412359	+	0.126911i	−0.502391	−	0.557961i	−0.139017	−	1.32266i	2.08726	+	1.51649i	−0.978148	−	0.207912i	0.0500950	+	0.0867670i
16.10	0.844998	−	0.938465i	0.104528	−	0.994522i	0.0423614	+	0.403042i	1.13050	−	3.47932i	−0.844998	−	0.938465i	−0.100820	−	0.959240i	2.45734	+	1.78536i	−0.978148	−	0.207912i	−2.30995	−	4.00095i
16.11	1.06574	−	1.18363i	0.104528	−	0.994522i	−0.0561096	−	0.533847i	−1.11978	+	3.44633i	−1.06574	−	1.18363i	0.311034	+	2.95930i	1.88542	+	1.36983i	−0.978148	−	0.207912i	2.88578	+	4.99832i
16.12	1.44079	−	1.60016i	0.104528	−	0.994522i	−0.275575	−	2.62192i	0.0727615	−	0.223937i	−1.44079	−	1.60016i	−0.0981000	−	0.933359i	−1.10854	−	0.805401i	−0.978148	−	0.207912i	−0.253500	−	0.439075i
16.13	1.65656	−	1.83979i	0.104528	−	0.994522i	−0.431601	−	4.10641i	−0.543715	+	1.67338i	−1.65656	−	1.83979i	−0.506656	−	4.82051i	−4.26418	−	3.09811i	−0.978148	−	0.207912i	2.17798	+	3.77238i
16.14	1.71487	−	1.90455i	0.104528	−	0.994522i	−0.477496	−	4.54307i	1.12068	−	3.44910i	−1.71487	−	1.90455i	0.446007	+	4.24348i	−5.32461	−	3.86856i	−0.978148	−	0.207912i	−4.64718	−	8.04915i
256.1	−0.273007	−	2.59749i	0.978148	+	0.207912i	−4.71612	+	1.00244i	0.822674	+	0.597708i	0.273007	−	2.59749i	3.34488	−	0.710976i	2.27719	+	7.00848i	0.913545	+	0.406737i	1.32794	−	2.30007i
256.2	−0.258546	−	2.45990i	0.978148	+	0.207912i	−4.02797	+	0.856172i	−0.872789	−	0.634118i	0.258546	−	2.45990i	−3.11026	+	0.661107i	1.61884	+	4.98227i	0.913545	+	0.406737i	−1.33421	+	2.31092i
256.3	−0.220383	−	2.09680i	0.978148	+	0.207912i	−2.39171	+	0.508373i	3.27922	+	2.38249i	0.220383	−	2.09680i	−0.946033	+	0.201086i	0.290016	+	0.892577i	0.913545	+	0.406737i	4.27293	−	7.40092i
256.4	−0.140589	−	1.33762i	0.978148	+	0.207912i	0.186846	−	0.0397154i	−0.768940	−	0.558668i	0.140589	−	1.33762i	3.61727	−	0.768874i	−0.910638	−	2.80265i	0.913545	+	0.406737i	−0.639178	+	1.10709i
256.5	−0.134448	−	1.27919i	0.978148	+	0.207912i	0.338055	−	0.0718558i	−1.25069	−	0.908681i	0.134448	−	1.27919i	1.33154	−	0.283029i	−0.932303	−	2.86933i	0.913545	+	0.406737i	−0.994219	+	1.72204i
256.6	−0.0790719	−	0.752319i	0.978148	+	0.207912i	1.39656	−	0.296849i	2.62007	+	1.90359i	0.0790719	−	0.752319i	−0.683673	+	0.145319i	−0.801274	−	2.46607i	0.913545	+	0.406737i	1.22494	−	2.12165i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
13.c	even	3	1	inner
143.q	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.bg.a	✓	112
11.c	even	5	1	inner	429.2.bg.a	✓	112
13.c	even	3	1	inner	429.2.bg.a	✓	112
143.q	even	15	1	inner	429.2.bg.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.bg.a	✓	112	1.a	even	1	1	trivial
429.2.bg.a	✓	112	11.c	even	5	1	inner
429.2.bg.a	✓	112	13.c	even	3	1	inner
429.2.bg.a	✓	112	143.q	even	15	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(20\!\cdots\!92\)\( T_{2}^{84} - 93223698381 T_{2}^{83} + \)\(32\!\cdots\!96\)\( T_{2}^{82} + 850388036538 T_{2}^{81} + \)\(29\!\cdots\!09\)\( T_{2}^{80} + \)\(37\!\cdots\!33\)\( T_{2}^{79} - \)\(24\!\cdots\!94\)\( T_{2}^{78} - \)\(40\!\cdots\!93\)\( T_{2}^{77} + \)\(84\!\cdots\!89\)\( T_{2}^{76} + \)\(42\!\cdots\!82\)\( T_{2}^{75} - \)\(17\!\cdots\!31\)\( T_{2}^{74} + \)\(80\!\cdots\!80\)\( T_{2}^{73} - \)\(14\!\cdots\!12\)\( T_{2}^{72} - \)\(55\!\cdots\!68\)\( T_{2}^{71} + \)\(79\!\cdots\!73\)\( T_{2}^{70} + \)\(21\!\cdots\!46\)\( T_{2}^{69} - \)\(16\!\cdots\!40\)\( T_{2}^{68} - \)\(12\!\cdots\!35\)\( T_{2}^{67} - \)\(34\!\cdots\!31\)\( T_{2}^{66} - \)\(41\!\cdots\!42\)\( T_{2}^{65} + \)\(31\!\cdots\!41\)\( T_{2}^{64} + \)\(24\!\cdots\!63\)\( T_{2}^{63} - \)\(79\!\cdots\!27\)\( T_{2}^{62} - \)\(48\!\cdots\!63\)\( T_{2}^{61} + \)\(50\!\cdots\!18\)\( T_{2}^{60} - \)\(26\!\cdots\!96\)\( T_{2}^{59} + \)\(32\!\cdots\!22\)\( T_{2}^{58} + \)\(46\!\cdots\!81\)\( T_{2}^{57} - \)\(14\!\cdots\!16\)\( T_{2}^{56} - \)\(17\!\cdots\!93\)\( T_{2}^{55} + \)\(25\!\cdots\!23\)\( T_{2}^{54} + \)\(36\!\cdots\!79\)\( T_{2}^{53} + \)\(36\!\cdots\!95\)\( T_{2}^{52} - \)\(21\!\cdots\!78\)\( T_{2}^{51} - \)\(13\!\cdots\!84\)\( T_{2}^{50} - \)\(11\!\cdots\!57\)\( T_{2}^{49} + \)\(30\!\cdots\!39\)\( T_{2}^{48} + \)\(34\!\cdots\!20\)\( T_{2}^{47} - \)\(27\!\cdots\!97\)\( T_{2}^{46} - \)\(39\!\cdots\!59\)\( T_{2}^{45} - \)\(16\!\cdots\!23\)\( T_{2}^{44} - \)\(77\!\cdots\!37\)\( T_{2}^{43} + \)\(10\!\cdots\!90\)\( T_{2}^{42} + \)\(10\!\cdots\!45\)\( T_{2}^{41} - \)\(18\!\cdots\!99\)\( T_{2}^{40} - \)\(17\!\cdots\!44\)\( T_{2}^{39} + \)\(13\!\cdots\!11\)\( T_{2}^{38} + \)\(83\!\cdots\!55\)\( T_{2}^{37} + \)\(16\!\cdots\!50\)\( T_{2}^{36} + \)\(21\!\cdots\!66\)\( T_{2}^{35} - \)\(45\!\cdots\!19\)\( T_{2}^{34} - \)\(43\!\cdots\!53\)\( T_{2}^{33} + \)\(34\!\cdots\!20\)\( T_{2}^{32} + \)\(26\!\cdots\!74\)\( T_{2}^{31} + \)\(63\!\cdots\!87\)\( T_{2}^{30} + \)\(17\!\cdots\!08\)\( T_{2}^{29} - \)\(15\!\cdots\!07\)\( T_{2}^{28} - \)\(25\!\cdots\!91\)\( T_{2}^{27} + \)\(96\!\cdots\!73\)\( T_{2}^{26} + \)\(13\!\cdots\!94\)\( T_{2}^{25} - \)\(14\!\cdots\!41\)\( T_{2}^{24} - \)\(34\!\cdots\!68\)\( T_{2}^{23} - \)\(19\!\cdots\!23\)\( T_{2}^{22} - \)\(69\!\cdots\!88\)\( T_{2}^{21} + \)\(26\!\cdots\!95\)\( T_{2}^{20} + \)\(20\!\cdots\!96\)\( T_{2}^{19} - \)\(13\!\cdots\!72\)\( T_{2}^{18} - \)\(10\!\cdots\!63\)\( T_{2}^{17} + \)\(30\!\cdots\!31\)\( T_{2}^{16} + \)\(14\!\cdots\!25\)\( T_{2}^{15} + \)\(65\!\cdots\!65\)\( T_{2}^{14} + \)\(67\!\cdots\!25\)\( T_{2}^{13} + \)\(15\!\cdots\!25\)\( T_{2}^{12} - \)\(79\!\cdots\!25\)\( T_{2}^{11} + \)\(24\!\cdots\!00\)\( T_{2}^{10} + \)\(16\!\cdots\!25\)\( T_{2}^{9} - \)\(22\!\cdots\!50\)\( T_{2}^{8} - \)\(65\!\cdots\!00\)\( T_{2}^{7} + \)\(74\!\cdots\!50\)\( T_{2}^{6} - \)\(82\!\cdots\!00\)\( T_{2}^{5} + \)\(46\!\cdots\!75\)\( T_{2}^{4} - \)\(36\!\cdots\!50\)\( T_{2}^{3} + \)\(15\!\cdots\!75\)\( T_{2}^{2} - \)\(14\!\cdots\!75\)\( T_{2} + 152587890625 \)">\(T_{2}^{112} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).