Properties

Label 966.2.y.a
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.

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

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 −0.740155 3.05096i 0.415415 0.909632i 0.385204 2.61756i 0.142315 0.989821i 0.723734 + 0.690079i 3.08272 0.594146i
25.2 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.678320 2.79607i 0.415415 0.909632i −0.304541 + 2.62817i 0.142315 0.989821i 0.723734 + 0.690079i 2.82518 0.544509i
25.3 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.582238 2.40002i 0.415415 0.909632i −0.0849128 + 2.64439i 0.142315 0.989821i 0.723734 + 0.690079i 2.42500 0.467381i
25.4 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.0760992 + 0.313685i 0.415415 0.909632i 2.64554 0.0334955i 0.142315 0.989821i 0.723734 + 0.690079i −0.316951 + 0.0610872i
25.5 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.373397 + 1.53917i 0.415415 0.909632i −2.64405 0.0947728i 0.142315 0.989821i 0.723734 + 0.690079i −1.55519 + 0.299738i
25.6 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.445036 + 1.83446i 0.415415 0.909632i 2.08936 1.62314i 0.142315 0.989821i 0.723734 + 0.690079i −1.85356 + 0.357245i
25.7 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.534809 + 2.20451i 0.415415 0.909632i −2.08203 1.63252i 0.142315 0.989821i 0.723734 + 0.690079i −2.22746 + 0.429308i
25.8 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.779910 + 3.21483i 0.415415 0.909632i 0.591550 + 2.57877i 0.142315 0.989821i 0.723734 + 0.690079i −3.24830 + 0.626059i
121.1 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.150385 3.15698i −0.654861 + 0.755750i −2.56837 + 0.635207i 0.959493 0.281733i −0.888835 + 0.458227i −2.48437 + 1.95373i
121.2 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.128045 2.68799i −0.654861 + 0.755750i 0.622376 + 2.57151i 0.959493 0.281733i −0.888835 + 0.458227i −2.11530 + 1.66349i
121.3 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.0861989 1.80954i −0.654861 + 0.755750i 1.04990 2.42852i 0.959493 0.281733i −0.888835 + 0.458227i −1.42401 + 1.11985i
121.4 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.00766190 + 0.160843i −0.654861 + 0.755750i −0.825321 + 2.51373i 0.959493 0.281733i −0.888835 + 0.458227i 0.126575 0.0995393i
121.5 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.0236187 + 0.495818i −0.654861 + 0.755750i 2.48735 0.901724i 0.959493 0.281733i −0.888835 + 0.458227i 0.390181 0.306842i
121.6 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.106503 + 2.23577i −0.654861 + 0.755750i −2.31212 1.28612i 0.959493 0.281733i −0.888835 + 0.458227i 1.75943 1.38363i
121.7 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.122200 + 2.56530i −0.654861 + 0.755750i −0.168303 + 2.64039i 0.959493 0.281733i −0.888835 + 0.458227i 2.01875 1.58756i
121.8 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.170819 + 3.58593i −0.654861 + 0.755750i 2.64569 + 0.0174095i 0.959493 0.281733i −0.888835 + 0.458227i 2.82193 2.21919i
151.1 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −4.11027 + 0.392484i −0.142315 0.989821i −2.51634 + 0.817338i −0.841254 + 0.540641i 0.580057 0.814576i −0.973442 + 4.01258i
151.2 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.84187 + 0.271366i −0.142315 0.989821i 2.39273 + 1.12909i −0.841254 + 0.540641i 0.580057 0.814576i −0.673045 + 2.77433i
151.3 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.66247 + 0.254235i −0.142315 0.989821i 1.24716 + 2.33337i −0.841254 + 0.540641i 0.580057 0.814576i −0.630556 + 2.59919i
151.4 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −1.00054 + 0.0955401i −0.142315 0.989821i −2.50267 0.858289i −0.841254 + 0.540641i 0.580057 0.814576i −0.236960 + 0.976761i
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.a 160
7.c even 3 1 inner 966.2.y.a 160
23.c even 11 1 inner 966.2.y.a 160
161.m even 33 1 inner 966.2.y.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.a 160 1.a even 1 1 trivial
966.2.y.a 160 7.c even 3 1 inner
966.2.y.a 160 23.c even 11 1 inner
966.2.y.a 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(24\!\cdots\!32\)\( T_{5}^{140} - \)\(18\!\cdots\!03\)\( T_{5}^{139} - \)\(26\!\cdots\!76\)\( T_{5}^{138} - \)\(81\!\cdots\!56\)\( T_{5}^{137} + \)\(10\!\cdots\!49\)\( T_{5}^{136} + \)\(64\!\cdots\!31\)\( T_{5}^{135} + \)\(16\!\cdots\!50\)\( T_{5}^{134} + \)\(14\!\cdots\!52\)\( T_{5}^{133} - \)\(54\!\cdots\!78\)\( T_{5}^{132} - \)\(31\!\cdots\!57\)\( T_{5}^{131} + \)\(33\!\cdots\!97\)\( T_{5}^{130} + \)\(19\!\cdots\!05\)\( T_{5}^{129} - \)\(28\!\cdots\!61\)\( T_{5}^{128} + \)\(25\!\cdots\!06\)\( T_{5}^{127} - \)\(30\!\cdots\!33\)\( T_{5}^{126} - \)\(52\!\cdots\!39\)\( T_{5}^{125} + \)\(63\!\cdots\!29\)\( T_{5}^{124} + \)\(21\!\cdots\!72\)\( T_{5}^{123} - \)\(82\!\cdots\!33\)\( T_{5}^{122} + \)\(29\!\cdots\!66\)\( T_{5}^{121} + \)\(28\!\cdots\!07\)\( T_{5}^{120} - \)\(64\!\cdots\!46\)\( T_{5}^{119} + \)\(86\!\cdots\!81\)\( T_{5}^{118} + \)\(26\!\cdots\!98\)\( T_{5}^{117} - \)\(10\!\cdots\!80\)\( T_{5}^{116} + \)\(41\!\cdots\!81\)\( T_{5}^{115} + \)\(37\!\cdots\!51\)\( T_{5}^{114} - \)\(46\!\cdots\!06\)\( T_{5}^{113} + \)\(13\!\cdots\!08\)\( T_{5}^{112} + \)\(42\!\cdots\!78\)\( T_{5}^{111} - \)\(76\!\cdots\!41\)\( T_{5}^{110} + \)\(20\!\cdots\!03\)\( T_{5}^{109} + \)\(47\!\cdots\!38\)\( T_{5}^{108} - \)\(17\!\cdots\!03\)\( T_{5}^{107} + \)\(10\!\cdots\!83\)\( T_{5}^{106} + \)\(24\!\cdots\!83\)\( T_{5}^{105} - \)\(67\!\cdots\!55\)\( T_{5}^{104} + \)\(93\!\cdots\!70\)\( T_{5}^{103} + \)\(39\!\cdots\!39\)\( T_{5}^{102} - \)\(71\!\cdots\!51\)\( T_{5}^{101} + \)\(39\!\cdots\!01\)\( T_{5}^{100} + \)\(73\!\cdots\!60\)\( T_{5}^{99} - \)\(31\!\cdots\!90\)\( T_{5}^{98} + \)\(35\!\cdots\!50\)\( T_{5}^{97} + \)\(19\!\cdots\!77\)\( T_{5}^{96} - \)\(41\!\cdots\!01\)\( T_{5}^{95} + \)\(49\!\cdots\!34\)\( T_{5}^{94} + \)\(17\!\cdots\!34\)\( T_{5}^{93} - \)\(94\!\cdots\!67\)\( T_{5}^{92} + \)\(70\!\cdots\!99\)\( T_{5}^{91} + \)\(53\!\cdots\!28\)\( T_{5}^{90} - \)\(10\!\cdots\!80\)\( T_{5}^{89} - \)\(60\!\cdots\!35\)\( T_{5}^{88} + \)\(40\!\cdots\!30\)\( T_{5}^{87} - \)\(18\!\cdots\!52\)\( T_{5}^{86} - \)\(14\!\cdots\!31\)\( T_{5}^{85} + \)\(93\!\cdots\!41\)\( T_{5}^{84} - \)\(13\!\cdots\!78\)\( T_{5}^{83} - \)\(28\!\cdots\!09\)\( T_{5}^{82} + \)\(10\!\cdots\!55\)\( T_{5}^{81} - \)\(14\!\cdots\!20\)\( T_{5}^{80} - \)\(21\!\cdots\!35\)\( T_{5}^{79} + \)\(20\!\cdots\!47\)\( T_{5}^{78} - \)\(84\!\cdots\!42\)\( T_{5}^{77} - \)\(55\!\cdots\!15\)\( T_{5}^{76} + \)\(11\!\cdots\!83\)\( T_{5}^{75} + \)\(17\!\cdots\!20\)\( T_{5}^{74} - \)\(10\!\cdots\!44\)\( T_{5}^{73} + \)\(19\!\cdots\!30\)\( T_{5}^{72} + \)\(20\!\cdots\!21\)\( T_{5}^{71} - \)\(10\!\cdots\!04\)\( T_{5}^{70} + \)\(46\!\cdots\!05\)\( T_{5}^{69} + \)\(51\!\cdots\!53\)\( T_{5}^{68} - \)\(12\!\cdots\!92\)\( T_{5}^{67} + \)\(72\!\cdots\!40\)\( T_{5}^{66} + \)\(36\!\cdots\!52\)\( T_{5}^{65} - \)\(82\!\cdots\!57\)\( T_{5}^{64} + \)\(18\!\cdots\!65\)\( T_{5}^{63} + \)\(24\!\cdots\!79\)\( T_{5}^{62} - \)\(37\!\cdots\!63\)\( T_{5}^{61} - \)\(13\!\cdots\!99\)\( T_{5}^{60} + \)\(12\!\cdots\!36\)\( T_{5}^{59} - \)\(13\!\cdots\!39\)\( T_{5}^{58} - \)\(17\!\cdots\!30\)\( T_{5}^{57} + \)\(53\!\cdots\!42\)\( T_{5}^{56} - \)\(23\!\cdots\!39\)\( T_{5}^{55} - \)\(84\!\cdots\!94\)\( T_{5}^{54} + \)\(11\!\cdots\!22\)\( T_{5}^{53} + \)\(88\!\cdots\!99\)\( T_{5}^{52} - \)\(30\!\cdots\!35\)\( T_{5}^{51} - \)\(10\!\cdots\!02\)\( T_{5}^{50} + \)\(86\!\cdots\!29\)\( T_{5}^{49} - \)\(43\!\cdots\!65\)\( T_{5}^{48} - \)\(15\!\cdots\!17\)\( T_{5}^{47} + \)\(26\!\cdots\!97\)\( T_{5}^{46} - \)\(59\!\cdots\!32\)\( T_{5}^{45} - \)\(24\!\cdots\!18\)\( T_{5}^{44} + \)\(28\!\cdots\!27\)\( T_{5}^{43} - \)\(19\!\cdots\!98\)\( T_{5}^{42} + \)\(47\!\cdots\!55\)\( T_{5}^{41} - \)\(21\!\cdots\!52\)\( T_{5}^{40} - \)\(13\!\cdots\!94\)\( T_{5}^{39} + \)\(17\!\cdots\!46\)\( T_{5}^{38} + \)\(52\!\cdots\!02\)\( T_{5}^{37} - \)\(24\!\cdots\!54\)\( T_{5}^{36} + \)\(90\!\cdots\!41\)\( T_{5}^{35} + \)\(15\!\cdots\!10\)\( T_{5}^{34} - \)\(10\!\cdots\!84\)\( T_{5}^{33} - \)\(32\!\cdots\!15\)\( T_{5}^{32} + \)\(55\!\cdots\!07\)\( T_{5}^{31} + \)\(10\!\cdots\!08\)\( T_{5}^{30} - \)\(52\!\cdots\!04\)\( T_{5}^{29} - \)\(86\!\cdots\!24\)\( T_{5}^{28} - \)\(21\!\cdots\!87\)\( T_{5}^{27} + \)\(74\!\cdots\!53\)\( T_{5}^{26} + \)\(12\!\cdots\!63\)\( T_{5}^{25} - \)\(25\!\cdots\!83\)\( T_{5}^{24} + \)\(54\!\cdots\!62\)\( T_{5}^{23} + \)\(98\!\cdots\!13\)\( T_{5}^{22} - \)\(30\!\cdots\!08\)\( T_{5}^{21} + \)\(18\!\cdots\!59\)\( T_{5}^{20} + \)\(66\!\cdots\!93\)\( T_{5}^{19} - \)\(41\!\cdots\!62\)\( T_{5}^{18} + \)\(15\!\cdots\!35\)\( T_{5}^{17} + \)\(13\!\cdots\!99\)\( T_{5}^{16} + \)\(55\!\cdots\!84\)\( T_{5}^{15} + \)\(26\!\cdots\!37\)\( T_{5}^{14} + \)\(18\!\cdots\!91\)\( T_{5}^{13} + \)\(39\!\cdots\!23\)\( T_{5}^{12} + \)\(19\!\cdots\!38\)\( T_{5}^{11} + \)\(91\!\cdots\!20\)\( T_{5}^{10} + \)\(21\!\cdots\!81\)\( T_{5}^{9} + \)\(70\!\cdots\!65\)\( T_{5}^{8} + \)\(25\!\cdots\!23\)\( T_{5}^{7} + \)\(56\!\cdots\!45\)\( T_{5}^{6} + \)\(12\!\cdots\!03\)\( T_{5}^{5} + \)\(29\!\cdots\!90\)\( T_{5}^{4} + \)\(46\!\cdots\!40\)\( T_{5}^{3} + \)\(61\!\cdots\!21\)\( T_{5}^{2} + \)\(72\!\cdots\!57\)\( T_{5} + \)\(47\!\cdots\!81\)\( \)">\(T_{5}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).