Newform orbit 930.2.bo.a

• Label: 930.2.bo.a
• Level: $930$
• Weight: $2$
• Character orbit: 930.bo
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bo (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 256q - 64q^{2} - 64q^{4} - 2q^{5} - 64q^{8} + 4q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	0.309017	+	0.951057i	−1.73157	−	0.0407018i	−0.809017	+	0.587785i	0.493029	+	2.18104i	−0.496376	−	1.65940i	−0.863747	+	1.94001i	−0.809017	−	0.587785i	2.99669	+	0.140956i	−1.92193	+	1.14288i
179.2	0.309017	+	0.951057i	−1.69787	−	0.342394i	−0.809017	+	0.587785i	−1.92997	−	1.12925i	−0.199035	−	1.72058i	−0.934663	+	2.09929i	−0.809017	−	0.587785i	2.76553	+	1.16268i	0.477584	−	2.18447i
179.3	0.309017	+	0.951057i	−1.66354	−	0.482315i	−0.809017	+	0.587785i	0.575623	−	2.16071i	−0.0553539	−	1.73117i	−1.66082	+	3.73026i	−0.809017	−	0.587785i	2.53474	+	1.60470i	2.23283	−	0.120245i
179.4	0.309017	+	0.951057i	−1.64568	+	0.540140i	−0.809017	+	0.587785i	2.22856	−	0.183115i	−1.02225	−	1.39822i	0.318388	−	0.715111i	−0.809017	−	0.587785i	2.41650	−	1.77779i	0.862815	+	2.06290i
179.5	0.309017	+	0.951057i	−1.59415	−	0.677259i	−0.809017	+	0.587785i	−2.09420	+	0.783790i	0.151491	−	1.72541i	1.79370	−	4.02872i	−0.809017	−	0.587785i	2.08264	+	2.15931i	−1.39257	−	1.74950i
179.6	0.309017	+	0.951057i	−1.46052	−	0.931061i	−0.809017	+	0.587785i	1.43804	+	1.71232i	0.434165	−	1.67675i	0.429946	−	0.965674i	−0.809017	−	0.587785i	1.26625	+	2.71967i	−1.18414	+	1.89679i
179.7	0.309017	+	0.951057i	−1.41936	+	0.992677i	−0.809017	+	0.587785i	−1.50750	+	1.65150i	−1.38270	−	1.04314i	0.732525	−	1.64528i	−0.809017	−	0.587785i	1.02919	−	2.81794i	−2.03651	−	0.923374i
179.8	0.309017	+	0.951057i	−1.39110	+	1.03191i	−0.809017	+	0.587785i	−1.53101	−	1.62973i	−1.41128	−	1.00414i	0.325961	−	0.732121i	−0.809017	−	0.587785i	0.870333	−	2.87098i	1.07685	−	1.95969i
179.9	0.309017	+	0.951057i	−1.21541	−	1.23401i	−0.809017	+	0.587785i	1.34040	−	1.78978i	0.798025	−	1.53726i	1.34591	−	3.02296i	−0.809017	−	0.587785i	−0.0455387	+	2.99965i	2.11639	+	0.721724i
179.10	0.309017	+	0.951057i	−1.03655	+	1.38764i	−0.809017	+	0.587785i	−1.97712	+	1.04451i	−1.64004	−	0.557016i	−1.83029	+	4.11091i	−0.809017	−	0.587785i	−0.851111	−	2.87674i	−1.60435	−	1.55758i
179.11	0.309017	+	0.951057i	−0.835073	−	1.51745i	−0.809017	+	0.587785i	2.23544	+	0.0527903i	1.18513	−	1.26312i	−0.915394	+	2.05601i	−0.809017	−	0.587785i	−1.60531	+	2.53436i	0.640584	+	2.14235i
179.12	0.309017	+	0.951057i	−0.732396	+	1.56958i	−0.809017	+	0.587785i	0.719639	+	2.11710i	−1.71909	−	0.211521i	1.13296	−	2.54466i	−0.809017	−	0.587785i	−1.92719	−	2.29911i	−1.79110	+	1.33864i
179.13	0.309017	+	0.951057i	−0.696767	+	1.58572i	−0.809017	+	0.587785i	1.79380	−	1.33503i	−1.72342	−	0.172650i	0.308507	−	0.692917i	−0.809017	−	0.587785i	−2.02903	−	2.20976i	1.82400	+	1.29346i
179.14	0.309017	+	0.951057i	−0.584760	−	1.63035i	−0.809017	+	0.587785i	−2.12123	+	0.707392i	1.36986	−	1.05995i	−1.37162	+	3.08072i	−0.809017	−	0.587785i	−2.31611	+	1.90673i	−1.32826	−	1.79881i
179.15	0.309017	+	0.951057i	−0.128921	−	1.72725i	−0.809017	+	0.587785i	0.447993	+	2.19073i	1.60287	−	0.656359i	1.37162	−	3.08072i	−0.809017	−	0.587785i	−2.96676	+	0.445355i	−1.94507	+	1.10304i
179.16	0.309017	+	0.951057i	0.0869307	+	1.72987i	−0.809017	+	0.587785i	−2.09670	−	0.777090i	−1.61834	+	0.617235i	1.84852	−	4.15185i	−0.809017	−	0.587785i	−2.98489	+	0.300757i	0.0911422	−	2.23421i
179.17	0.309017	+	0.951057i	0.145674	−	1.72591i	−0.809017	+	0.587785i	−1.16344	−	1.90956i	1.68646	−	0.394792i	0.915394	−	2.05601i	−0.809017	−	0.587785i	−2.95756	−	0.502843i	1.45657	−	1.69658i
179.18	0.309017	+	0.951057i	0.198005	+	1.72070i	−0.809017	+	0.587785i	2.18990	−	0.452039i	−1.57529	+	0.720038i	0.478693	−	1.07516i	−0.809017	−	0.587785i	−2.92159	+	0.681412i	1.10663	+	1.94303i
179.19	0.309017	+	0.951057i	0.518984	+	1.65247i	−0.809017	+	0.587785i	−0.703473	−	2.12253i	−1.41122	+	1.00422i	−0.478693	+	1.07516i	−0.809017	−	0.587785i	−2.46131	+	1.71521i	1.80126	−	1.32494i
179.20	0.309017	+	0.951057i	0.608421	−	1.62167i	−0.809017	+	0.587785i	0.879797	−	2.05571i	1.73032	+	0.0775177i	−1.34591	+	3.02296i	−0.809017	−	0.587785i	−2.25965	−	1.97332i	2.22697	+	0.201486i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	inner
31.h	odd	30	1	inner
465.bm	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bo.a	✓	256
3.b	odd	2	1		930.2.bo.b	yes	256
5.b	even	2	1		930.2.bo.b	yes	256
15.d	odd	2	1	inner	930.2.bo.a	✓	256
31.h	odd	30	1	inner	930.2.bo.a	✓	256
93.p	even	30	1		930.2.bo.b	yes	256
155.v	odd	30	1		930.2.bo.b	yes	256
465.bm	even	30	1	inner	930.2.bo.a	✓	256

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bo.a	✓	256	1.a	even	1	1	trivial
930.2.bo.a	✓	256	15.d	odd	2	1	inner
930.2.bo.a	✓	256	31.h	odd	30	1	inner
930.2.bo.a	✓	256	465.bm	even	30	1	inner
930.2.bo.b	yes	256	3.b	odd	2	1
930.2.bo.b	yes	256	5.b	even	2	1
930.2.bo.b	yes	256	93.p	even	30	1
930.2.bo.b	yes	256	155.v	odd	30	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(13\!\cdots\!09\)\( T_{17}^{114} + \)\(42\!\cdots\!81\)\( T_{17}^{113} - \)\(49\!\cdots\!13\)\( T_{17}^{112} + \)\(26\!\cdots\!39\)\( T_{17}^{111} - \)\(14\!\cdots\!41\)\( T_{17}^{110} + \)\(62\!\cdots\!96\)\( T_{17}^{109} + \)\(87\!\cdots\!13\)\( T_{17}^{108} - \)\(11\!\cdots\!70\)\( T_{17}^{107} + \)\(51\!\cdots\!21\)\( T_{17}^{106} - \)\(14\!\cdots\!49\)\( T_{17}^{105} + \)\(14\!\cdots\!33\)\( T_{17}^{104} - \)\(50\!\cdots\!37\)\( T_{17}^{103} + \)\(14\!\cdots\!78\)\( T_{17}^{102} - \)\(62\!\cdots\!55\)\( T_{17}^{101} - \)\(16\!\cdots\!76\)\( T_{17}^{100} + \)\(24\!\cdots\!15\)\( T_{17}^{99} - \)\(81\!\cdots\!12\)\( T_{17}^{98} + \)\(15\!\cdots\!31\)\( T_{17}^{97} - \)\(19\!\cdots\!44\)\( T_{17}^{96} + \)\(36\!\cdots\!45\)\( T_{17}^{95} - \)\(10\!\cdots\!15\)\( T_{17}^{94} + \)\(10\!\cdots\!02\)\( T_{17}^{93} + \)\(13\!\cdots\!48\)\( T_{17}^{92} - \)\(20\!\cdots\!99\)\( T_{17}^{91} + \)\(67\!\cdots\!18\)\( T_{17}^{90} - \)\(52\!\cdots\!14\)\( T_{17}^{89} + \)\(18\!\cdots\!85\)\( T_{17}^{88} + \)\(46\!\cdots\!12\)\( T_{17}^{87} + \)\(31\!\cdots\!69\)\( T_{17}^{86} + \)\(75\!\cdots\!19\)\( T_{17}^{85} + \)\(24\!\cdots\!36\)\( T_{17}^{84} + \)\(29\!\cdots\!58\)\( T_{17}^{83} - \)\(47\!\cdots\!63\)\( T_{17}^{82} + \)\(68\!\cdots\!16\)\( T_{17}^{81} - \)\(20\!\cdots\!18\)\( T_{17}^{80} + \)\(97\!\cdots\!48\)\( T_{17}^{79} - \)\(31\!\cdots\!78\)\( T_{17}^{78} + \)\(26\!\cdots\!22\)\( T_{17}^{77} - \)\(70\!\cdots\!34\)\( T_{17}^{76} - \)\(25\!\cdots\!50\)\( T_{17}^{75} + \)\(90\!\cdots\!29\)\( T_{17}^{74} - \)\(76\!\cdots\!97\)\( T_{17}^{73} + \)\(25\!\cdots\!52\)\( T_{17}^{72} - \)\(12\!\cdots\!94\)\( T_{17}^{71} + \)\(41\!\cdots\!42\)\( T_{17}^{70} - \)\(13\!\cdots\!61\)\( T_{17}^{69} + \)\(42\!\cdots\!34\)\( T_{17}^{68} - \)\(53\!\cdots\!38\)\( T_{17}^{67} + \)\(15\!\cdots\!38\)\( T_{17}^{66} + \)\(10\!\cdots\!79\)\( T_{17}^{65} - \)\(34\!\cdots\!18\)\( T_{17}^{64} + \)\(26\!\cdots\!98\)\( T_{17}^{63} - \)\(75\!\cdots\!79\)\( T_{17}^{62} + \)\(28\!\cdots\!26\)\( T_{17}^{61} - \)\(81\!\cdots\!45\)\( T_{17}^{60} + \)\(14\!\cdots\!95\)\( T_{17}^{59} - \)\(41\!\cdots\!25\)\( T_{17}^{58} - \)\(91\!\cdots\!39\)\( T_{17}^{57} + \)\(28\!\cdots\!58\)\( T_{17}^{56} - \)\(29\!\cdots\!71\)\( T_{17}^{55} + \)\(86\!\cdots\!31\)\( T_{17}^{54} - \)\(31\!\cdots\!82\)\( T_{17}^{53} + \)\(84\!\cdots\!04\)\( T_{17}^{52} - \)\(14\!\cdots\!13\)\( T_{17}^{51} + \)\(37\!\cdots\!44\)\( T_{17}^{50} + \)\(51\!\cdots\!74\)\( T_{17}^{49} - \)\(13\!\cdots\!88\)\( T_{17}^{48} + \)\(16\!\cdots\!89\)\( T_{17}^{47} - \)\(51\!\cdots\!06\)\( T_{17}^{46} + \)\(17\!\cdots\!14\)\( T_{17}^{45} - \)\(46\!\cdots\!57\)\( T_{17}^{44} + \)\(64\!\cdots\!66\)\( T_{17}^{43} - \)\(58\!\cdots\!92\)\( T_{17}^{42} - \)\(21\!\cdots\!22\)\( T_{17}^{41} + \)\(12\!\cdots\!59\)\( T_{17}^{40} - \)\(59\!\cdots\!57\)\( T_{17}^{39} + \)\(18\!\cdots\!15\)\( T_{17}^{38} - \)\(43\!\cdots\!33\)\( T_{17}^{37} + \)\(90\!\cdots\!41\)\( T_{17}^{36} - \)\(15\!\cdots\!30\)\( T_{17}^{35} + \)\(10\!\cdots\!46\)\( T_{17}^{34} + \)\(10\!\cdots\!03\)\( T_{17}^{33} - \)\(51\!\cdots\!04\)\( T_{17}^{32} + \)\(13\!\cdots\!48\)\( T_{17}^{31} - \)\(24\!\cdots\!88\)\( T_{17}^{30} + \)\(23\!\cdots\!73\)\( T_{17}^{29} + \)\(17\!\cdots\!02\)\( T_{17}^{28} - \)\(14\!\cdots\!38\)\( T_{17}^{27} + \)\(42\!\cdots\!18\)\( T_{17}^{26} - \)\(86\!\cdots\!61\)\( T_{17}^{25} + \)\(14\!\cdots\!17\)\( T_{17}^{24} - \)\(19\!\cdots\!91\)\( T_{17}^{23} + \)\(20\!\cdots\!90\)\( T_{17}^{22} - \)\(19\!\cdots\!99\)\( T_{17}^{21} + \)\(15\!\cdots\!10\)\( T_{17}^{20} - \)\(91\!\cdots\!67\)\( T_{17}^{19} + \)\(44\!\cdots\!56\)\( T_{17}^{18} - \)\(19\!\cdots\!91\)\( T_{17}^{17} + \)\(55\!\cdots\!49\)\( T_{17}^{16} - \)\(68\!\cdots\!04\)\( T_{17}^{15} + \)\(30\!\cdots\!42\)\( T_{17}^{14} - \)\(49\!\cdots\!41\)\( T_{17}^{13} - \)\(36\!\cdots\!71\)\( T_{17}^{12} - \)\(25\!\cdots\!55\)\( T_{17}^{11} - \)\(29\!\cdots\!55\)\( T_{17}^{10} + \)\(41\!\cdots\!02\)\( T_{17}^{9} + \)\(15\!\cdots\!81\)\( T_{17}^{8} + \)\(25\!\cdots\!36\)\( T_{17}^{7} + \)\(28\!\cdots\!52\)\( T_{17}^{6} + \)\(23\!\cdots\!12\)\( T_{17}^{5} + \)\(13\!\cdots\!32\)\( T_{17}^{4} + \)\(50\!\cdots\!88\)\( T_{17}^{3} + \)\(12\!\cdots\!84\)\( T_{17}^{2} + \)\(18\!\cdots\!64\)\( T_{17} + \)\(12\!\cdots\!36\)\( \)">\(T_{17}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).