Properties

Label 966.2.y.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 8q^{2} - 8q^{3} + 8q^{4} + 2q^{5} + 16q^{6} - 2q^{7} - 16q^{8} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q + 8q^{2} - 8q^{3} + 8q^{4} + 2q^{5} + 16q^{6} - 2q^{7} - 16q^{8} + 8q^{9} + 2q^{10} + 8q^{11} - 8q^{12} - 4q^{13} + 9q^{14} - 18q^{15} + 8q^{16} - 21q^{17} + 8q^{18} + 4q^{19} + 18q^{20} - 15q^{21} + 116q^{22} + 14q^{23} + 80q^{24} + 2q^{25} + 2q^{26} + 16q^{27} - 18q^{28} - 46q^{29} - 2q^{30} + q^{31} + 8q^{32} - 8q^{33} - 68q^{34} - 84q^{35} - 16q^{36} + 19q^{37} - 7q^{38} - 2q^{39} - 9q^{40} + 38q^{41} + 2q^{42} + 28q^{43} + 8q^{44} + 24q^{45} - 8q^{46} - 26q^{47} + 16q^{48} - 6q^{49} - 4q^{50} + 21q^{51} + 2q^{52} - 20q^{53} - 8q^{54} + 46q^{55} - 2q^{56} - 14q^{57} - 21q^{58} + 36q^{59} - 2q^{60} + 40q^{61} - 24q^{62} - 2q^{63} - 16q^{64} - 10q^{65} - 8q^{66} + 6q^{67} + 34q^{68} + 6q^{69} - 20q^{70} + 132q^{71} + 8q^{72} - 36q^{73} + 19q^{74} - 2q^{75} - 8q^{76} - 75q^{77} - 18q^{78} + 34q^{79} + 2q^{80} + 8q^{81} + 47q^{82} + 74q^{83} + 13q^{84} + 2q^{85} + 30q^{86} + 10q^{87} - 3q^{88} - 12q^{89} - 4q^{90} - 30q^{91} - 50q^{92} + 10q^{93} - 4q^{94} + 27q^{95} - 8q^{96} + 24q^{97} + 7q^{98} - 16q^{99} + O(q^{100}) \)

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 −0.835382 3.44349i −0.415415 + 0.909632i 0.912603 2.48338i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.47934 + 0.670588i
25.2 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.748398 3.08494i −0.415415 + 0.909632i 2.13670 + 1.56029i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.11705 + 0.600763i
25.3 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.228734 0.942856i −0.415415 + 0.909632i −1.88597 + 1.85556i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.952672 + 0.183612i
25.4 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.211998 0.873870i −0.415415 + 0.909632i 2.39932 + 1.11502i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.882967 + 0.170178i
25.5 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.167516 0.690511i −0.415415 + 0.909632i −2.01446 1.71521i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.697700 + 0.134471i
25.6 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.244232 + 1.00674i −0.415415 + 0.909632i 0.601440 2.57648i −0.142315 + 0.989821i 0.723734 + 0.690079i 1.01722 0.196053i
25.7 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.751462 + 3.09757i −0.415415 + 0.909632i 2.38356 1.14832i −0.142315 + 0.989821i 0.723734 + 0.690079i 3.12982 0.603223i
25.8 0.0475819 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.853588 + 3.51854i −0.415415 + 0.909632i −1.36069 + 2.26904i −0.142315 + 0.989821i 0.723734 + 0.690079i 3.55517 0.685202i
121.1 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.163478 3.43182i 0.654861 0.755750i 0.873534 2.49739i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.70065 2.12382i
121.2 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.161073 3.38134i 0.654861 0.755750i 2.36003 + 1.19594i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.66093 2.09258i
121.3 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.142165 2.98442i 0.654861 0.755750i −2.19584 + 1.47590i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.34857 1.84694i
121.4 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.0167727 0.352103i 0.654861 0.755750i −1.88460 + 1.85696i −0.959493 + 0.281733i −0.888835 + 0.458227i 0.277085 0.217902i
121.5 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.00337646 0.0708807i 0.654861 0.755750i 1.92801 + 1.81184i −0.959493 + 0.281733i −0.888835 + 0.458227i 0.0557792 0.0438652i
121.6 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.00889795 + 0.186791i 0.654861 0.755750i 2.27500 1.35069i −0.959493 + 0.281733i −0.888835 + 0.458227i −0.146994 + 0.115597i
121.7 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.0874971 + 1.83679i 0.654861 0.755750i 0.484830 2.60095i −0.959493 + 0.281733i −0.888835 + 0.458227i −1.44545 + 1.13672i
121.8 0.580057 + 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.141678 + 2.97420i 0.654861 0.755750i −2.12708 1.57338i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.34053 + 1.84061i
151.1 −0.327068 + 0.945001i 0.888835 0.458227i −0.786053 0.618159i −3.08451 + 0.294535i 0.142315 + 0.989821i −0.562151 + 2.58534i 0.841254 0.540641i 0.580057 0.814576i 0.730508 3.01119i
151.2 −0.327068 + 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.78783 + 0.266206i 0.142315 + 0.989821i 2.46897 0.950880i 0.841254 0.540641i 0.580057 0.814576i 0.660246 2.72157i
151.3 −0.327068 + 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.45803 + 0.234713i 0.142315 + 0.989821i −0.472531 2.60321i 0.841254 0.540641i 0.580057 0.814576i 0.582139 2.39961i
151.4 −0.327068 + 0.945001i 0.888835 0.458227i −0.786053 0.618159i −0.150887 + 0.0144079i 0.142315 + 0.989821i −2.59084 + 0.536227i 0.841254 0.540641i 0.580057 0.814576i 0.0357347 0.147300i
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.c 160
7.c even 3 1 inner 966.2.y.c 160
23.c even 11 1 inner 966.2.y.c 160
161.m even 33 1 inner 966.2.y.c 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.c 160 1.a even 1 1 trivial
966.2.y.c 160 7.c even 3 1 inner
966.2.y.c 160 23.c even 11 1 inner
966.2.y.c 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(13\!\cdots\!09\)\( T_{5}^{139} + \)\(26\!\cdots\!42\)\( T_{5}^{138} - \)\(18\!\cdots\!88\)\( T_{5}^{137} + \)\(49\!\cdots\!49\)\( T_{5}^{136} + \)\(22\!\cdots\!99\)\( T_{5}^{135} - \)\(13\!\cdots\!72\)\( T_{5}^{134} - \)\(40\!\cdots\!96\)\( T_{5}^{133} + \)\(13\!\cdots\!38\)\( T_{5}^{132} - \)\(29\!\cdots\!05\)\( T_{5}^{131} + \)\(13\!\cdots\!53\)\( T_{5}^{130} + \)\(41\!\cdots\!99\)\( T_{5}^{129} - \)\(13\!\cdots\!31\)\( T_{5}^{128} - \)\(24\!\cdots\!30\)\( T_{5}^{127} + \)\(20\!\cdots\!67\)\( T_{5}^{126} - \)\(21\!\cdots\!33\)\( T_{5}^{125} - \)\(12\!\cdots\!69\)\( T_{5}^{124} + \)\(60\!\cdots\!96\)\( T_{5}^{123} - \)\(84\!\cdots\!17\)\( T_{5}^{122} - \)\(44\!\cdots\!06\)\( T_{5}^{121} + \)\(22\!\cdots\!81\)\( T_{5}^{120} - \)\(85\!\cdots\!52\)\( T_{5}^{119} - \)\(16\!\cdots\!81\)\( T_{5}^{118} + \)\(51\!\cdots\!20\)\( T_{5}^{117} - \)\(35\!\cdots\!66\)\( T_{5}^{116} - \)\(43\!\cdots\!37\)\( T_{5}^{115} + \)\(18\!\cdots\!39\)\( T_{5}^{114} - \)\(87\!\cdots\!90\)\( T_{5}^{113} - \)\(15\!\cdots\!68\)\( T_{5}^{112} + \)\(31\!\cdots\!24\)\( T_{5}^{111} + \)\(20\!\cdots\!61\)\( T_{5}^{110} - \)\(27\!\cdots\!09\)\( T_{5}^{109} + \)\(83\!\cdots\!14\)\( T_{5}^{108} + \)\(44\!\cdots\!39\)\( T_{5}^{107} - \)\(73\!\cdots\!85\)\( T_{5}^{106} + \)\(92\!\cdots\!75\)\( T_{5}^{105} + \)\(11\!\cdots\!75\)\( T_{5}^{104} - \)\(80\!\cdots\!12\)\( T_{5}^{103} + \)\(12\!\cdots\!67\)\( T_{5}^{102} + \)\(37\!\cdots\!33\)\( T_{5}^{101} - \)\(31\!\cdots\!87\)\( T_{5}^{100} - \)\(64\!\cdots\!80\)\( T_{5}^{99} - \)\(32\!\cdots\!76\)\( T_{5}^{98} + \)\(23\!\cdots\!66\)\( T_{5}^{97} + \)\(35\!\cdots\!43\)\( T_{5}^{96} - \)\(14\!\cdots\!43\)\( T_{5}^{95} - \)\(27\!\cdots\!20\)\( T_{5}^{94} + \)\(64\!\cdots\!00\)\( T_{5}^{93} + \)\(20\!\cdots\!73\)\( T_{5}^{92} + \)\(61\!\cdots\!31\)\( T_{5}^{91} + \)\(86\!\cdots\!82\)\( T_{5}^{90} - \)\(41\!\cdots\!22\)\( T_{5}^{89} - \)\(37\!\cdots\!03\)\( T_{5}^{88} + \)\(19\!\cdots\!50\)\( T_{5}^{87} + \)\(29\!\cdots\!92\)\( T_{5}^{86} - \)\(41\!\cdots\!91\)\( T_{5}^{85} - \)\(10\!\cdots\!57\)\( T_{5}^{84} + \)\(42\!\cdots\!02\)\( T_{5}^{83} + \)\(46\!\cdots\!17\)\( T_{5}^{82} - \)\(26\!\cdots\!87\)\( T_{5}^{81} - \)\(48\!\cdots\!30\)\( T_{5}^{80} + \)\(15\!\cdots\!79\)\( T_{5}^{79} + \)\(37\!\cdots\!95\)\( T_{5}^{78} - \)\(10\!\cdots\!30\)\( T_{5}^{77} - \)\(25\!\cdots\!69\)\( T_{5}^{76} + \)\(48\!\cdots\!53\)\( T_{5}^{75} + \)\(17\!\cdots\!86\)\( T_{5}^{74} - \)\(11\!\cdots\!60\)\( T_{5}^{73} - \)\(99\!\cdots\!54\)\( T_{5}^{72} - \)\(38\!\cdots\!87\)\( T_{5}^{71} + \)\(33\!\cdots\!98\)\( T_{5}^{70} + \)\(52\!\cdots\!75\)\( T_{5}^{69} - \)\(20\!\cdots\!17\)\( T_{5}^{68} - \)\(23\!\cdots\!86\)\( T_{5}^{67} - \)\(40\!\cdots\!82\)\( T_{5}^{66} + \)\(28\!\cdots\!98\)\( T_{5}^{65} + \)\(20\!\cdots\!31\)\( T_{5}^{64} + \)\(17\!\cdots\!77\)\( T_{5}^{63} - \)\(40\!\cdots\!77\)\( T_{5}^{62} - \)\(96\!\cdots\!83\)\( T_{5}^{61} - \)\(58\!\cdots\!01\)\( T_{5}^{60} + \)\(20\!\cdots\!26\)\( T_{5}^{59} + \)\(22\!\cdots\!01\)\( T_{5}^{58} - \)\(20\!\cdots\!84\)\( T_{5}^{57} - \)\(50\!\cdots\!86\)\( T_{5}^{56} - \)\(12\!\cdots\!03\)\( T_{5}^{55} + \)\(59\!\cdots\!40\)\( T_{5}^{54} + \)\(57\!\cdots\!48\)\( T_{5}^{53} - \)\(18\!\cdots\!37\)\( T_{5}^{52} - \)\(78\!\cdots\!97\)\( T_{5}^{51} - \)\(43\!\cdots\!42\)\( T_{5}^{50} + \)\(26\!\cdots\!91\)\( T_{5}^{49} + \)\(78\!\cdots\!73\)\( T_{5}^{48} + \)\(54\!\cdots\!85\)\( T_{5}^{47} - \)\(28\!\cdots\!35\)\( T_{5}^{46} - \)\(78\!\cdots\!96\)\( T_{5}^{45} - \)\(70\!\cdots\!88\)\( T_{5}^{44} + \)\(15\!\cdots\!21\)\( T_{5}^{43} + \)\(11\!\cdots\!32\)\( T_{5}^{42} + \)\(63\!\cdots\!11\)\( T_{5}^{41} - \)\(67\!\cdots\!68\)\( T_{5}^{40} - \)\(13\!\cdots\!90\)\( T_{5}^{39} - \)\(43\!\cdots\!64\)\( T_{5}^{38} + \)\(66\!\cdots\!16\)\( T_{5}^{37} + \)\(80\!\cdots\!62\)\( T_{5}^{36} + \)\(22\!\cdots\!17\)\( T_{5}^{35} - \)\(45\!\cdots\!32\)\( T_{5}^{34} - \)\(24\!\cdots\!22\)\( T_{5}^{33} + \)\(21\!\cdots\!99\)\( T_{5}^{32} + \)\(32\!\cdots\!53\)\( T_{5}^{31} + \)\(16\!\cdots\!76\)\( T_{5}^{30} + \)\(89\!\cdots\!68\)\( T_{5}^{29} - \)\(29\!\cdots\!08\)\( T_{5}^{28} - \)\(18\!\cdots\!65\)\( T_{5}^{27} + \)\(13\!\cdots\!39\)\( T_{5}^{26} + \)\(72\!\cdots\!21\)\( T_{5}^{25} + \)\(46\!\cdots\!47\)\( T_{5}^{24} - \)\(97\!\cdots\!64\)\( T_{5}^{23} - \)\(68\!\cdots\!47\)\( T_{5}^{22} - \)\(38\!\cdots\!94\)\( T_{5}^{21} - \)\(66\!\cdots\!35\)\( T_{5}^{20} + \)\(30\!\cdots\!73\)\( T_{5}^{19} + \)\(39\!\cdots\!10\)\( T_{5}^{18} + \)\(20\!\cdots\!59\)\( T_{5}^{17} + \)\(12\!\cdots\!59\)\( T_{5}^{16} - \)\(78\!\cdots\!72\)\( T_{5}^{15} - \)\(16\!\cdots\!69\)\( T_{5}^{14} - \)\(83\!\cdots\!31\)\( T_{5}^{13} + \)\(87\!\cdots\!39\)\( T_{5}^{12} + \)\(32\!\cdots\!42\)\( T_{5}^{11} + \)\(52\!\cdots\!22\)\( T_{5}^{10} + \)\(11\!\cdots\!75\)\( T_{5}^{9} + \)\(20\!\cdots\!95\)\( T_{5}^{8} + \)\(13\!\cdots\!45\)\( T_{5}^{7} + \)\(30\!\cdots\!99\)\( T_{5}^{6} + \)\(53\!\cdots\!57\)\( T_{5}^{5} - \)\(11\!\cdots\!20\)\( T_{5}^{4} - \)\(16\!\cdots\!90\)\( T_{5}^{3} + \)\(20\!\cdots\!11\)\( T_{5}^{2} - \)\(17\!\cdots\!59\)\( T_{5} + \)\(18\!\cdots\!81\)\( \)">\(T_{5}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).