Properties

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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 160q - 8q^{2} + 8q^{3} + 8q^{4} + 6q^{5} + 16q^{6} - 10q^{7} + 16q^{8} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q - 8q^{2} + 8q^{3} + 8q^{4} + 6q^{5} + 16q^{6} - 10q^{7} + 16q^{8} + 8q^{9} - 6q^{10} + 8q^{11} + 8q^{12} + 4q^{13} - 13q^{14} + 10q^{15} + 8q^{16} + 9q^{17} - 8q^{18} - 8q^{19} + 10q^{20} - 3q^{21} - 28q^{22} + 22q^{23} + 80q^{24} + 46q^{25} + 2q^{26} - 16q^{27} - 14q^{28} + 62q^{29} - 6q^{30} + 7q^{31} - 8q^{32} - 14q^{33} - 4q^{34} + 24q^{35} - 16q^{36} + 15q^{37} + 19q^{38} - 2q^{39} + 5q^{40} - 14q^{41} + 10q^{42} - 56q^{43} + 8q^{44} - 16q^{45} + 66q^{47} - 16q^{48} + 42q^{49} + 4q^{50} + 9q^{51} - 2q^{52} + 12q^{53} - 8q^{54} - 14q^{55} + 10q^{56} - 6q^{57} - 13q^{58} + 62q^{59} + 6q^{60} + 16q^{61} - 8q^{62} + 2q^{63} - 16q^{64} + 70q^{65} - 8q^{66} - 14q^{67} - 46q^{68} + 22q^{69} + 24q^{70} - 108q^{71} - 8q^{72} + 40q^{73} - 15q^{74} + 2q^{75} + 16q^{76} + 29q^{77} + 18q^{78} + 22q^{79} + 6q^{80} + 8q^{81} - 7q^{82} - 18q^{83} - 9q^{84} + 26q^{85} - 6q^{86} + 2q^{87} + 3q^{88} + 112q^{89} + 12q^{90} + 18q^{91} - 66q^{92} - 26q^{93} + 23q^{95} - 8q^{96} + 8q^{97} + 9q^{98} + 28q^{99} + O(q^{100}) \)

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
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.8
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(966, [\chi])\).