Newform orbit 966.2.y.b

• Label: 966.2.y.b
• Level: $966$
• Weight: $2$
• Character orbit: 966.y
• Analytic conductor: $7.714$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.71354883526\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160q - 8q^{2} + 8q^{3} + 8q^{4} + 6q^{5} + 16q^{6} - 10q^{7} + 16q^{8} + 8q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
25.1	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	−0.495624	−	2.04299i	−0.415415	+	0.909632i	−2.46554	+	0.959758i	0.142315	−	0.989821i	0.723734	+	0.690079i	2.06426	−	0.397853i
25.2	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	−0.494800	−	2.03959i	−0.415415	+	0.909632i	2.60154	+	0.481642i	0.142315	−	0.989821i	0.723734	+	0.690079i	2.06083	−	0.397192i
25.3	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	−0.365274	−	1.50568i	−0.415415	+	0.909632i	0.990231	+	2.45346i	0.142315	−	0.989821i	0.723734	+	0.690079i	1.52136	−	0.293217i
25.4	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	0.0842779	+	0.347398i	−0.415415	+	0.909632i	0.863387	−	2.50091i	0.142315	−	0.989821i	0.723734	+	0.690079i	−0.351015	+	0.0676526i
25.5	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	0.149969	+	0.618179i	−0.415415	+	0.909632i	−1.95013	+	1.78801i	0.142315	−	0.989821i	0.723734	+	0.690079i	−0.624615	+	0.120385i
25.6	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	0.152691	+	0.629402i	−0.415415	+	0.909632i	−1.69725	−	2.02961i	0.142315	−	0.989821i	0.723734	+	0.690079i	−0.635954	+	0.122570i
25.7	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	0.666702	+	2.74818i	−0.415415	+	0.909632i	1.08121	+	2.41474i	0.142315	−	0.989821i	0.723734	+	0.690079i	−2.77679	+	0.535183i
25.8	−0.0475819	+	0.998867i	0.928368	+	0.371662i	−0.995472	−	0.0950560i	0.902348	+	3.71953i	−0.415415	+	0.909632i	1.54544	−	2.14747i	0.142315	−	0.989821i	0.723734	+	0.690079i	−3.75825	+	0.724344i
121.1	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	−0.204977	−	4.30300i	0.654861	−	0.755750i	−0.384103	−	2.61772i	0.959493	−	0.281733i	−0.888835	+	0.458227i	−3.38622	+	2.66296i
121.2	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	−0.116162	−	2.43853i	0.654861	−	0.755750i	2.62255	−	0.349625i	0.959493	−	0.281733i	−0.888835	+	0.458227i	−1.91899	+	1.50911i
121.3	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	−0.0822082	−	1.72576i	0.654861	−	0.755750i	−1.65083	+	2.06755i	0.959493	−	0.281733i	−0.888835	+	0.458227i	−1.35808	+	1.06801i
121.4	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	−0.0296258	−	0.621921i	0.654861	−	0.755750i	−2.62091	−	0.361724i	0.959493	−	0.281733i	−0.888835	+	0.458227i	−0.489417	+	0.384882i
121.5	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	0.0297398	+	0.624314i	0.654861	−	0.755750i	2.47488	−	0.935389i	0.959493	−	0.281733i	−0.888835	+	0.458227i	0.491301	−	0.386363i
121.6	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	0.0618096	+	1.29754i	0.654861	−	0.755750i	−1.05626	−	2.42576i	0.959493	−	0.281733i	−0.888835	+	0.458227i	1.02109	−	0.802998i
121.7	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	0.121343	+	2.54731i	0.654861	−	0.755750i	0.313232	+	2.62714i	0.959493	−	0.281733i	−0.888835	+	0.458227i	2.00459	−	1.57643i
121.8	−0.580057	−	0.814576i	0.235759	+	0.971812i	−0.327068	+	0.945001i	0.161616	+	3.39274i	0.654861	−	0.755750i	1.77884	+	1.95851i	0.959493	−	0.281733i	−0.888835	+	0.458227i	2.66990	−	2.09963i
151.1	0.327068	−	0.945001i	−0.888835	+	0.458227i	−0.786053	−	0.618159i	−3.90496	+	0.372879i	0.142315	+	0.989821i	2.64365	+	0.105392i	−0.841254	+	0.540641i	0.580057	−	0.814576i	−0.924818	+	3.81215i
151.2	0.327068	−	0.945001i	−0.888835	+	0.458227i	−0.786053	−	0.618159i	−3.73412	+	0.356565i	0.142315	+	0.989821i	−2.47167	−	0.943859i	−0.841254	+	0.540641i	0.580057	−	0.814576i	−0.884357	+	3.64537i
151.3	0.327068	−	0.945001i	−0.888835	+	0.458227i	−0.786053	−	0.618159i	−1.62160	+	0.154844i	0.142315	+	0.989821i	−1.14236	+	2.38642i	−0.841254	+	0.540641i	0.580057	−	0.814576i	−0.384046	+	1.58306i
151.4	0.327068	−	0.945001i	−0.888835	+	0.458227i	−0.786053	−	0.618159i	−1.13751	+	0.108619i	0.142315	+	0.989821i	1.15912	+	2.37833i	−0.841254	+	0.540641i	0.580057	−	0.814576i	−0.269397	+	1.11047i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
23.c	even	11	1	inner
161.m	even	33	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	966.2.y.b	✓	160
7.c	even	3	1	inner	966.2.y.b	✓	160
23.c	even	11	1	inner	966.2.y.b	✓	160
161.m	even	33	1	inner	966.2.y.b	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.y.b	✓	160	1.a	even	1	1	trivial
966.2.y.b	✓	160	7.c	even	3	1	inner
966.2.y.b	✓	160	23.c	even	11	1	inner
966.2.y.b	✓	160	161.m	even	33	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!77\)\( T_{5}^{139} - \)\(43\!\cdots\!24\)\( T_{5}^{138} - \)\(17\!\cdots\!24\)\( T_{5}^{137} + \)\(10\!\cdots\!85\)\( T_{5}^{136} + \)\(12\!\cdots\!53\)\( T_{5}^{135} - \)\(17\!\cdots\!10\)\( T_{5}^{134} + \)\(10\!\cdots\!64\)\( T_{5}^{133} + \)\(18\!\cdots\!88\)\( T_{5}^{132} - \)\(47\!\cdots\!03\)\( T_{5}^{131} - \)\(79\!\cdots\!75\)\( T_{5}^{130} + \)\(77\!\cdots\!67\)\( T_{5}^{129} - \)\(11\!\cdots\!67\)\( T_{5}^{128} - \)\(79\!\cdots\!78\)\( T_{5}^{127} + \)\(28\!\cdots\!91\)\( T_{5}^{126} + \)\(50\!\cdots\!81\)\( T_{5}^{125} - \)\(33\!\cdots\!23\)\( T_{5}^{124} - \)\(55\!\cdots\!88\)\( T_{5}^{123} + \)\(24\!\cdots\!57\)\( T_{5}^{122} - \)\(31\!\cdots\!28\)\( T_{5}^{121} - \)\(91\!\cdots\!33\)\( T_{5}^{120} + \)\(46\!\cdots\!56\)\( T_{5}^{119} - \)\(12\!\cdots\!79\)\( T_{5}^{118} - \)\(38\!\cdots\!54\)\( T_{5}^{117} + \)\(36\!\cdots\!56\)\( T_{5}^{116} + \)\(19\!\cdots\!73\)\( T_{5}^{115} - \)\(15\!\cdots\!73\)\( T_{5}^{114} - \)\(14\!\cdots\!56\)\( T_{5}^{113} - \)\(93\!\cdots\!78\)\( T_{5}^{112} - \)\(77\!\cdots\!06\)\( T_{5}^{111} - \)\(12\!\cdots\!87\)\( T_{5}^{110} + \)\(70\!\cdots\!23\)\( T_{5}^{109} + \)\(81\!\cdots\!62\)\( T_{5}^{108} - \)\(19\!\cdots\!59\)\( T_{5}^{107} - \)\(70\!\cdots\!13\)\( T_{5}^{106} - \)\(14\!\cdots\!33\)\( T_{5}^{105} + \)\(41\!\cdots\!61\)\( T_{5}^{104} + \)\(14\!\cdots\!78\)\( T_{5}^{103} + \)\(36\!\cdots\!27\)\( T_{5}^{102} + \)\(56\!\cdots\!45\)\( T_{5}^{101} - \)\(24\!\cdots\!29\)\( T_{5}^{100} - \)\(90\!\cdots\!28\)\( T_{5}^{99} - \)\(46\!\cdots\!78\)\( T_{5}^{98} + \)\(59\!\cdots\!62\)\( T_{5}^{97} + \)\(16\!\cdots\!05\)\( T_{5}^{96} + \)\(16\!\cdots\!25\)\( T_{5}^{95} - \)\(93\!\cdots\!80\)\( T_{5}^{94} - \)\(25\!\cdots\!72\)\( T_{5}^{93} - \)\(41\!\cdots\!03\)\( T_{5}^{92} + \)\(14\!\cdots\!97\)\( T_{5}^{91} + \)\(34\!\cdots\!00\)\( T_{5}^{90} - \)\(72\!\cdots\!30\)\( T_{5}^{89} - \)\(16\!\cdots\!87\)\( T_{5}^{88} - \)\(39\!\cdots\!68\)\( T_{5}^{87} - \)\(29\!\cdots\!80\)\( T_{5}^{86} + \)\(13\!\cdots\!13\)\( T_{5}^{85} + \)\(58\!\cdots\!55\)\( T_{5}^{84} + \)\(88\!\cdots\!94\)\( T_{5}^{83} - \)\(13\!\cdots\!35\)\( T_{5}^{82} - \)\(96\!\cdots\!57\)\( T_{5}^{81} - \)\(13\!\cdots\!34\)\( T_{5}^{80} + \)\(27\!\cdots\!51\)\( T_{5}^{79} + \)\(12\!\cdots\!69\)\( T_{5}^{78} + \)\(10\!\cdots\!08\)\( T_{5}^{77} - \)\(40\!\cdots\!99\)\( T_{5}^{76} - \)\(11\!\cdots\!95\)\( T_{5}^{75} - \)\(24\!\cdots\!84\)\( T_{5}^{74} + \)\(39\!\cdots\!16\)\( T_{5}^{73} + \)\(68\!\cdots\!24\)\( T_{5}^{72} - \)\(23\!\cdots\!39\)\( T_{5}^{71} - \)\(22\!\cdots\!78\)\( T_{5}^{70} - \)\(26\!\cdots\!25\)\( T_{5}^{69} + \)\(18\!\cdots\!83\)\( T_{5}^{68} + \)\(77\!\cdots\!20\)\( T_{5}^{67} + \)\(56\!\cdots\!94\)\( T_{5}^{66} - \)\(27\!\cdots\!48\)\( T_{5}^{65} - \)\(70\!\cdots\!55\)\( T_{5}^{64} + \)\(42\!\cdots\!95\)\( T_{5}^{63} - \)\(31\!\cdots\!05\)\( T_{5}^{62} - \)\(98\!\cdots\!63\)\( T_{5}^{61} - \)\(10\!\cdots\!73\)\( T_{5}^{60} + \)\(67\!\cdots\!64\)\( T_{5}^{59} + \)\(37\!\cdots\!47\)\( T_{5}^{58} + \)\(44\!\cdots\!10\)\( T_{5}^{57} + \)\(76\!\cdots\!48\)\( T_{5}^{56} + \)\(17\!\cdots\!21\)\( T_{5}^{55} - \)\(48\!\cdots\!24\)\( T_{5}^{54} - \)\(95\!\cdots\!98\)\( T_{5}^{53} - \)\(15\!\cdots\!33\)\( T_{5}^{52} + \)\(12\!\cdots\!63\)\( T_{5}^{51} + \)\(34\!\cdots\!44\)\( T_{5}^{50} + \)\(54\!\cdots\!21\)\( T_{5}^{49} + \)\(30\!\cdots\!13\)\( T_{5}^{48} - \)\(34\!\cdots\!15\)\( T_{5}^{47} - \)\(75\!\cdots\!91\)\( T_{5}^{46} - \)\(90\!\cdots\!78\)\( T_{5}^{45} - \)\(17\!\cdots\!58\)\( T_{5}^{44} - \)\(22\!\cdots\!49\)\( T_{5}^{43} - \)\(88\!\cdots\!46\)\( T_{5}^{42} - \)\(36\!\cdots\!37\)\( T_{5}^{41} + \)\(10\!\cdots\!66\)\( T_{5}^{40} + \)\(94\!\cdots\!50\)\( T_{5}^{39} + \)\(10\!\cdots\!90\)\( T_{5}^{38} + \)\(17\!\cdots\!66\)\( T_{5}^{37} + \)\(16\!\cdots\!18\)\( T_{5}^{36} + \)\(24\!\cdots\!33\)\( T_{5}^{35} + \)\(53\!\cdots\!38\)\( T_{5}^{34} + \)\(13\!\cdots\!32\)\( T_{5}^{33} + \)\(31\!\cdots\!97\)\( T_{5}^{32} - \)\(17\!\cdots\!35\)\( T_{5}^{31} + \)\(43\!\cdots\!92\)\( T_{5}^{30} + \)\(24\!\cdots\!16\)\( T_{5}^{29} - \)\(38\!\cdots\!32\)\( T_{5}^{28} + \)\(28\!\cdots\!15\)\( T_{5}^{27} + \)\(13\!\cdots\!19\)\( T_{5}^{26} - \)\(39\!\cdots\!63\)\( T_{5}^{25} - \)\(21\!\cdots\!93\)\( T_{5}^{24} + \)\(58\!\cdots\!54\)\( T_{5}^{23} - \)\(25\!\cdots\!21\)\( T_{5}^{22} + \)\(31\!\cdots\!16\)\( T_{5}^{21} + \)\(17\!\cdots\!43\)\( T_{5}^{20} - \)\(10\!\cdots\!51\)\( T_{5}^{19} + \)\(31\!\cdots\!98\)\( T_{5}^{18} + \)\(33\!\cdots\!57\)\( T_{5}^{17} - \)\(28\!\cdots\!73\)\( T_{5}^{16} + \)\(11\!\cdots\!96\)\( T_{5}^{15} + \)\(32\!\cdots\!69\)\( T_{5}^{14} - \)\(50\!\cdots\!93\)\( T_{5}^{13} + \)\(27\!\cdots\!03\)\( T_{5}^{12} - \)\(20\!\cdots\!84\)\( T_{5}^{11} - \)\(52\!\cdots\!14\)\( T_{5}^{10} + \)\(41\!\cdots\!85\)\( T_{5}^{9} - \)\(14\!\cdots\!73\)\( T_{5}^{8} + \)\(93\!\cdots\!27\)\( T_{5}^{7} + \)\(21\!\cdots\!77\)\( T_{5}^{6} - \)\(16\!\cdots\!47\)\( T_{5}^{5} + \)\(74\!\cdots\!00\)\( T_{5}^{4} - \)\(25\!\cdots\!30\)\( T_{5}^{3} + \)\(66\!\cdots\!91\)\( T_{5}^{2} - \)\(12\!\cdots\!13\)\( T_{5} + \)\(13\!\cdots\!61\)\( \)">\(T_{5}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).