Newform orbit 429.2.bg.b

• Label: 429.2.bg.b
• 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.

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

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	−1.60050	+	1.77754i	−0.104528	+	0.994522i	−0.388975	−	3.70085i	−1.06190	+	3.26819i	−1.60050	−	1.77754i	0.204434	+	1.94506i	3.33076	+	2.41994i	−0.978148	−	0.207912i	−4.10976	−	7.11831i
16.2	−1.53199	+	1.70145i	−0.104528	+	0.994522i	−0.338874	−	3.22417i	0.485300	−	1.49360i	−1.53199	−	1.70145i	0.0971269	+	0.924101i	2.30039	+	1.67133i	−0.978148	−	0.207912i	1.79781	+	3.11390i
16.3	−1.22841	+	1.36429i	−0.104528	+	0.994522i	−0.143231	−	1.36275i	−0.972545	+	2.99319i	−1.22841	−	1.36429i	−0.369323	−	3.51387i	−0.935298	−	0.679534i	−0.978148	−	0.207912i	−2.88888	−	5.00368i
16.4	−1.12855	+	1.25338i	−0.104528	+	0.994522i	−0.0882831	−	0.839957i	0.636771	−	1.95978i	−1.12855	−	1.25338i	−0.222872	−	2.12049i	−1.57654	−	1.14542i	−0.978148	−	0.207912i	1.73772	+	3.00982i
16.5	−0.802761	+	0.891557i	−0.104528	+	0.994522i	0.0586092	+	0.557629i	0.833954	−	2.56665i	−0.802761	−	0.891557i	0.259398	+	2.46800i	−2.48538	−	1.80573i	−0.978148	−	0.207912i	1.61885	+	2.80392i
16.6	−0.471127	+	0.523240i	−0.104528	+	0.994522i	0.157238	+	1.49602i	−0.263224	+	0.810122i	−0.471127	−	0.523240i	0.496767	+	4.72642i	−1.99610	−	1.45025i	−0.978148	−	0.207912i	−0.299876	−	0.519400i
16.7	0.0112957	−	0.0125452i	−0.104528	+	0.994522i	0.209027	+	1.98876i	1.04169	−	3.20599i	0.0112957	+	0.0125452i	−0.462856	−	4.40378i	0.0546248	+	0.0396872i	−0.978148	−	0.207912i	−0.0284531	−	0.0492821i
16.8	0.0581605	−	0.0645938i	−0.104528	+	0.994522i	0.208267	+	1.98153i	−0.913448	+	2.81130i	0.0581605	+	0.0645938i	−0.0279312	−	0.265748i	0.280746	+	0.203974i	−0.978148	−	0.207912i	0.128466	+	0.222510i
16.9	0.433437	−	0.481381i	−0.104528	+	0.994522i	0.165197	+	1.57175i	−0.543246	+	1.67194i	0.433437	+	0.481381i	0.0903479	+	0.859603i	1.87631	+	1.36322i	−0.978148	−	0.207912i	0.569377	+	0.986190i
16.10	0.682919	−	0.758458i	−0.104528	+	0.994522i	0.100176	+	0.953113i	0.0526100	−	0.161917i	0.682919	+	0.758458i	−0.171511	−	1.63181i	2.44268	+	1.77471i	−0.978148	−	0.207912i	−0.0868789	−	0.150479i
16.11	0.932419	−	1.03556i	−0.104528	+	0.994522i	0.00608545	+	0.0578992i	0.554150	−	1.70550i	0.932419	+	1.03556i	0.357902	+	3.40521i	2.32033	+	1.68582i	−0.978148	−	0.207912i	−1.24944	−	2.16409i
16.12	1.38006	−	1.53272i	−0.104528	+	0.994522i	−0.235585	−	2.24144i	−1.05420	+	3.24451i	1.38006	+	1.53272i	0.100634	+	0.957465i	−0.423465	−	0.307666i	−0.978148	−	0.207912i	3.51804	+	6.09342i
16.13	1.49760	−	1.66326i	−0.104528	+	0.994522i	−0.314550	−	2.99275i	0.954688	−	2.93823i	1.49760	+	1.66326i	−0.278334	−	2.64817i	−1.82740	−	1.32769i	−0.978148	−	0.207912i	−3.45728	−	5.98819i
16.14	1.76744	−	1.96294i	−0.104528	+	0.994522i	−0.520238	−	4.94974i	0.0584231	−	0.179808i	1.76744	+	1.96294i	−0.0737832	−	0.702000i	−6.36167	−	4.62202i	−0.978148	−	0.207912i	−0.249693	−	0.432481i
256.1	−0.265449	−	2.52558i	−0.978148	−	0.207912i	−4.35177	+	0.924997i	−0.205708	−	0.149455i	−0.265449	+	2.52558i	−0.893413	+	0.189901i	1.92183	+	5.91479i	0.913545	+	0.406737i	−0.322856	+	0.559203i
256.2	−0.245821	−	2.33883i	−0.978148	−	0.207912i	−3.45339	+	0.734041i	1.76043	+	1.27903i	−0.245821	+	2.33883i	2.44940	−	0.520637i	1.11227	+	3.42322i	0.913545	+	0.406737i	2.55867	−	4.43175i
256.3	−0.190759	−	1.81495i	−0.978148	−	0.207912i	−1.30136	+	0.276613i	−3.21129	−	2.33314i	−0.190759	+	1.81495i	3.99809	−	0.849821i	−0.377595	−	1.16212i	0.913545	+	0.406737i	−3.62195	+	6.27341i
256.4	−0.176325	−	1.67762i	−0.978148	−	0.207912i	−0.827032	+	0.175791i	0.993113	+	0.721539i	−0.176325	+	1.67762i	−1.11199	+	0.236360i	−0.601801	−	1.85215i	0.913545	+	0.406737i	1.03536	−	1.79330i
256.5	−0.121483	−	1.15584i	−0.978148	−	0.207912i	0.635096	−	0.134994i	−2.64351	−	1.92062i	−0.121483	+	1.15584i	−2.84394	+	0.604498i	−0.951465	−	2.92831i	0.913545	+	0.406737i	−1.89878	+	3.28879i
256.6	−0.0891258	−	0.847975i	−0.978148	−	0.207912i	1.24518	−	0.264670i	1.84578	+	1.34104i	−0.0891258	+	0.847975i	−2.53940	+	0.539767i	−0.862376	−	2.65412i	0.913545	+	0.406737i	0.972662	−	1.68470i

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.b	✓	112
11.c	even	5	1	inner	429.2.bg.b	✓	112
13.c	even	3	1	inner	429.2.bg.b	✓	112
143.q	even	15	1	inner	429.2.bg.b	✓	112

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(59\!\cdots\!70\)\( T_{2}^{82} + 673362357978 T_{2}^{81} + \)\(28\!\cdots\!43\)\( T_{2}^{80} - \)\(43\!\cdots\!01\)\( T_{2}^{79} - \)\(39\!\cdots\!18\)\( T_{2}^{78} + \)\(13\!\cdots\!73\)\( T_{2}^{77} - \)\(32\!\cdots\!51\)\( T_{2}^{76} + \)\(13\!\cdots\!96\)\( T_{2}^{75} + \)\(23\!\cdots\!49\)\( T_{2}^{74} - \)\(12\!\cdots\!52\)\( T_{2}^{73} - \)\(74\!\cdots\!02\)\( T_{2}^{72} - \)\(12\!\cdots\!52\)\( T_{2}^{71} + \)\(42\!\cdots\!83\)\( T_{2}^{70} + \)\(10\!\cdots\!88\)\( T_{2}^{69} + \)\(77\!\cdots\!84\)\( T_{2}^{68} - \)\(22\!\cdots\!47\)\( T_{2}^{67} - \)\(41\!\cdots\!59\)\( T_{2}^{66} - \)\(42\!\cdots\!32\)\( T_{2}^{65} + \)\(93\!\cdots\!53\)\( T_{2}^{64} + \)\(39\!\cdots\!37\)\( T_{2}^{63} + \)\(47\!\cdots\!65\)\( T_{2}^{62} - \)\(11\!\cdots\!73\)\( T_{2}^{61} - \)\(93\!\cdots\!26\)\( T_{2}^{60} + \)\(39\!\cdots\!06\)\( T_{2}^{59} + \)\(29\!\cdots\!00\)\( T_{2}^{58} + \)\(63\!\cdots\!35\)\( T_{2}^{57} - \)\(39\!\cdots\!46\)\( T_{2}^{56} - \)\(21\!\cdots\!19\)\( T_{2}^{55} - \)\(29\!\cdots\!23\)\( T_{2}^{54} + \)\(31\!\cdots\!07\)\( T_{2}^{53} + \)\(33\!\cdots\!83\)\( T_{2}^{52} - \)\(10\!\cdots\!60\)\( T_{2}^{51} - \)\(85\!\cdots\!64\)\( T_{2}^{50} + \)\(31\!\cdots\!35\)\( T_{2}^{49} + \)\(10\!\cdots\!85\)\( T_{2}^{48} - \)\(23\!\cdots\!32\)\( T_{2}^{47} - \)\(25\!\cdots\!63\)\( T_{2}^{46} - \)\(50\!\cdots\!63\)\( T_{2}^{45} - \)\(10\!\cdots\!19\)\( T_{2}^{44} + \)\(11\!\cdots\!91\)\( T_{2}^{43} + \)\(26\!\cdots\!16\)\( T_{2}^{42} - \)\(20\!\cdots\!97\)\( T_{2}^{41} - \)\(24\!\cdots\!27\)\( T_{2}^{40} + \)\(26\!\cdots\!68\)\( T_{2}^{39} - \)\(49\!\cdots\!17\)\( T_{2}^{38} - \)\(10\!\cdots\!65\)\( T_{2}^{37} + \)\(31\!\cdots\!52\)\( T_{2}^{36} - \)\(20\!\cdots\!40\)\( T_{2}^{35} - \)\(24\!\cdots\!27\)\( T_{2}^{34} + \)\(28\!\cdots\!61\)\( T_{2}^{33} + \)\(62\!\cdots\!06\)\( T_{2}^{32} - \)\(81\!\cdots\!50\)\( T_{2}^{31} + \)\(12\!\cdots\!03\)\( T_{2}^{30} - \)\(86\!\cdots\!64\)\( T_{2}^{29} - \)\(80\!\cdots\!71\)\( T_{2}^{28} + \)\(79\!\cdots\!49\)\( T_{2}^{27} + \)\(19\!\cdots\!53\)\( T_{2}^{26} - \)\(24\!\cdots\!54\)\( T_{2}^{25} + \)\(12\!\cdots\!29\)\( T_{2}^{24} - \)\(22\!\cdots\!60\)\( T_{2}^{23} - \)\(12\!\cdots\!69\)\( T_{2}^{22} + \)\(57\!\cdots\!14\)\( T_{2}^{21} + \)\(72\!\cdots\!37\)\( T_{2}^{20} - \)\(49\!\cdots\!86\)\( T_{2}^{19} - \)\(17\!\cdots\!00\)\( T_{2}^{18} + \)\(17\!\cdots\!91\)\( T_{2}^{17} - \)\(98\!\cdots\!41\)\( T_{2}^{16} + \)\(12\!\cdots\!83\)\( T_{2}^{15} + \)\(82\!\cdots\!33\)\( T_{2}^{14} - \)\(10\!\cdots\!03\)\( T_{2}^{13} + \)\(23\!\cdots\!65\)\( T_{2}^{12} + \)\(40\!\cdots\!89\)\( T_{2}^{11} - \)\(32\!\cdots\!18\)\( T_{2}^{10} + \)\(14\!\cdots\!51\)\( T_{2}^{9} + \)\(23\!\cdots\!60\)\( T_{2}^{8} + \)\(12\!\cdots\!02\)\( T_{2}^{7} + 33801426268 T_{2}^{6} + 953593498 T_{2}^{5} + 20268973 T_{2}^{4} + 155018 T_{2}^{3} + 2355 T_{2}^{2} + 99 T_{2} + 1 \)">\(T_{2}^{112} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).