Properties

Label 966.2.y.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 8q^{2} + 8q^{3} + 8q^{4} - 2q^{5} - 16q^{6} + 6q^{7} - 16q^{8} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q + 8q^{2} + 8q^{3} + 8q^{4} - 2q^{5} - 16q^{6} + 6q^{7} - 16q^{8} + 8q^{9} - 2q^{10} - 4q^{11} + 8q^{12} - 4q^{13} + 5q^{14} + 26q^{15} + 8q^{16} - 13q^{17} + 8q^{18} + 4q^{19} + 26q^{20} + 11q^{21} - 124q^{22} + 26q^{23} - 80q^{24} + 14q^{25} + 2q^{26} - 16q^{27} - 22q^{28} + 54q^{29} - 2q^{30} + 15q^{31} + 8q^{32} - 4q^{33} + 4q^{34} - 2q^{35} - 16q^{36} + 7q^{37} - 7q^{38} + 2q^{39} + 9q^{40} - 30q^{41} + 6q^{42} + 36q^{43} - 4q^{44} + 20q^{45} + 4q^{46} - 6q^{47} - 16q^{48} + 14q^{49} - 28q^{50} + 9q^{51} + 2q^{52} - 24q^{53} + 8q^{54} - 14q^{55} + 6q^{56} - 30q^{57} - 5q^{58} + 4q^{59} - 2q^{60} + 32q^{61} - 8q^{62} - 6q^{63} - 16q^{64} + 38q^{65} - 4q^{66} + 6q^{67} + 42q^{68} - 30q^{69} - 8q^{70} - 180q^{71} + 8q^{72} - 32q^{73} + 7q^{74} + 14q^{75} - 8q^{76} + 61q^{77} + 18q^{78} - 66q^{79} - 2q^{80} + 8q^{81} - 7q^{82} + 130q^{83} + 5q^{84} - 70q^{85} - 40q^{86} + 6q^{87} - 15q^{88} - 4q^{89} + 4q^{90} - 6q^{91} + 58q^{92} - 18q^{93} + 16q^{94} + 19q^{95} + 8q^{96} + 40q^{97} + 43q^{98} + 8q^{99} + O(q^{100}) \)

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 −0.786833 3.24337i 0.415415 0.909632i −2.25109 1.39018i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.27714 + 0.631616i
25.2 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.781948 3.22324i 0.415415 0.909632i 2.33065 1.25222i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.25679 + 0.627695i
25.3 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.363961 1.50027i 0.415415 0.909632i −0.661406 + 2.56175i −0.142315 + 0.989821i 0.723734 + 0.690079i −1.51589 + 0.292163i
25.4 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.167957 0.692328i 0.415415 0.909632i 2.14841 1.54413i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.699536 + 0.134824i
25.5 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.130603 0.538354i 0.415415 0.909632i −2.29097 + 1.32342i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.543959 + 0.104840i
25.6 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.112949 + 0.465581i 0.415415 0.909632i 1.20408 + 2.35588i −0.142315 + 0.989821i 0.723734 + 0.690079i 0.470428 0.0906676i
25.7 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.415021 + 1.71074i 0.415415 0.909632i −0.322897 2.62597i −0.142315 + 0.989821i 0.723734 + 0.690079i 1.72855 0.333151i
25.8 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.968835 + 3.99359i 0.415415 0.909632i 2.64295 0.121825i −0.142315 + 0.989821i 0.723734 + 0.690079i 4.03517 0.777715i
121.1 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.182773 3.83688i −0.654861 + 0.755750i −1.41671 + 2.23449i −0.959493 + 0.281733i −0.888835 + 0.458227i 3.01941 2.37449i
121.2 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.180250 3.78390i −0.654861 + 0.755750i 1.94204 + 1.79680i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.97772 2.34171i
121.3 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0817448 1.71603i −0.654861 + 0.755750i −2.45748 0.980189i −0.959493 + 0.281733i −0.888835 + 0.458227i 1.35042 1.06199i
121.4 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0809367 1.69907i −0.654861 + 0.755750i 1.60083 2.10650i −0.959493 + 0.281733i −0.888835 + 0.458227i 1.33707 1.05149i
121.5 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0478985 + 1.00551i −0.654861 + 0.755750i −0.794478 2.52365i −0.959493 + 0.281733i −0.888835 + 0.458227i −0.791284 + 0.622272i
121.6 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0802489 + 1.68463i −0.654861 + 0.755750i 1.50657 + 2.17491i −0.959493 + 0.281733i −0.888835 + 0.458227i −1.32571 + 1.04255i
121.7 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.132422 + 2.77988i −0.654861 + 0.755750i −1.83001 + 1.91077i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.18761 + 1.72035i
121.8 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.140981 + 2.95955i −0.654861 + 0.755750i 2.61692 0.389498i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.32900 + 1.83155i
151.1 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −4.06127 + 0.387805i −0.142315 0.989821i −0.441487 2.60866i 0.841254 0.540641i 0.580057 0.814576i 0.961837 3.96475i
151.2 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −2.76706 + 0.264222i −0.142315 0.989821i 2.35671 + 1.20246i 0.841254 0.540641i 0.580057 0.814576i 0.655327 2.70130i
151.3 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −0.688242 + 0.0657192i −0.142315 0.989821i −0.529942 + 2.59213i 0.841254 0.540641i 0.580057 0.814576i 0.162997 0.671884i
151.4 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −0.527443 + 0.0503647i −0.142315 0.989821i −2.62119 0.359704i 0.841254 0.540641i 0.580057 0.814576i 0.124915 0.514906i
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.d 160
7.c even 3 1 inner 966.2.y.d 160
23.c even 11 1 inner 966.2.y.d 160
161.m even 33 1 inner 966.2.y.d 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.d 160 1.a even 1 1 trivial
966.2.y.d 160 7.c even 3 1 inner
966.2.y.d 160 23.c even 11 1 inner
966.2.y.d 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(18\!\cdots\!56\)\( T_{5}^{140} + \)\(12\!\cdots\!21\)\( T_{5}^{139} + \)\(78\!\cdots\!54\)\( T_{5}^{138} + \)\(11\!\cdots\!08\)\( T_{5}^{137} - \)\(12\!\cdots\!03\)\( T_{5}^{136} - \)\(38\!\cdots\!83\)\( T_{5}^{135} + \)\(68\!\cdots\!08\)\( T_{5}^{134} + \)\(79\!\cdots\!44\)\( T_{5}^{133} + \)\(15\!\cdots\!84\)\( T_{5}^{132} - \)\(81\!\cdots\!15\)\( T_{5}^{131} - \)\(48\!\cdots\!47\)\( T_{5}^{130} - \)\(31\!\cdots\!03\)\( T_{5}^{129} + \)\(55\!\cdots\!79\)\( T_{5}^{128} + \)\(21\!\cdots\!18\)\( T_{5}^{127} + \)\(42\!\cdots\!83\)\( T_{5}^{126} - \)\(25\!\cdots\!01\)\( T_{5}^{125} - \)\(86\!\cdots\!89\)\( T_{5}^{124} + \)\(18\!\cdots\!56\)\( T_{5}^{123} + \)\(10\!\cdots\!01\)\( T_{5}^{122} + \)\(34\!\cdots\!28\)\( T_{5}^{121} + \)\(18\!\cdots\!89\)\( T_{5}^{120} - \)\(28\!\cdots\!54\)\( T_{5}^{119} - \)\(13\!\cdots\!21\)\( T_{5}^{118} - \)\(17\!\cdots\!84\)\( T_{5}^{117} + \)\(71\!\cdots\!54\)\( T_{5}^{116} + \)\(42\!\cdots\!03\)\( T_{5}^{115} + \)\(93\!\cdots\!31\)\( T_{5}^{114} - \)\(58\!\cdots\!12\)\( T_{5}^{113} - \)\(13\!\cdots\!34\)\( T_{5}^{112} - \)\(48\!\cdots\!16\)\( T_{5}^{111} - \)\(68\!\cdots\!73\)\( T_{5}^{110} + \)\(12\!\cdots\!55\)\( T_{5}^{109} + \)\(11\!\cdots\!62\)\( T_{5}^{108} + \)\(40\!\cdots\!83\)\( T_{5}^{107} + \)\(62\!\cdots\!23\)\( T_{5}^{106} - \)\(58\!\cdots\!57\)\( T_{5}^{105} - \)\(69\!\cdots\!69\)\( T_{5}^{104} - \)\(29\!\cdots\!36\)\( T_{5}^{103} - \)\(69\!\cdots\!73\)\( T_{5}^{102} - \)\(23\!\cdots\!31\)\( T_{5}^{101} + \)\(46\!\cdots\!27\)\( T_{5}^{100} + \)\(25\!\cdots\!56\)\( T_{5}^{99} + \)\(85\!\cdots\!88\)\( T_{5}^{98} + \)\(17\!\cdots\!54\)\( T_{5}^{97} + \)\(11\!\cdots\!99\)\( T_{5}^{96} - \)\(61\!\cdots\!49\)\( T_{5}^{95} - \)\(33\!\cdots\!86\)\( T_{5}^{94} - \)\(89\!\cdots\!94\)\( T_{5}^{93} - \)\(11\!\cdots\!51\)\( T_{5}^{92} + \)\(57\!\cdots\!93\)\( T_{5}^{91} + \)\(88\!\cdots\!98\)\( T_{5}^{90} + \)\(26\!\cdots\!56\)\( T_{5}^{89} + \)\(35\!\cdots\!13\)\( T_{5}^{88} + \)\(16\!\cdots\!12\)\( T_{5}^{87} - \)\(56\!\cdots\!56\)\( T_{5}^{86} - \)\(17\!\cdots\!15\)\( T_{5}^{85} + \)\(44\!\cdots\!73\)\( T_{5}^{84} + \)\(10\!\cdots\!10\)\( T_{5}^{83} + \)\(25\!\cdots\!35\)\( T_{5}^{82} + \)\(40\!\cdots\!45\)\( T_{5}^{81} + \)\(19\!\cdots\!88\)\( T_{5}^{80} - \)\(63\!\cdots\!87\)\( T_{5}^{79} - \)\(80\!\cdots\!47\)\( T_{5}^{78} + \)\(16\!\cdots\!48\)\( T_{5}^{77} + \)\(80\!\cdots\!59\)\( T_{5}^{76} + \)\(16\!\cdots\!75\)\( T_{5}^{75} + \)\(10\!\cdots\!82\)\( T_{5}^{74} - \)\(22\!\cdots\!32\)\( T_{5}^{73} - \)\(45\!\cdots\!64\)\( T_{5}^{72} + \)\(95\!\cdots\!65\)\( T_{5}^{71} + \)\(24\!\cdots\!52\)\( T_{5}^{70} + \)\(62\!\cdots\!89\)\( T_{5}^{69} + \)\(64\!\cdots\!77\)\( T_{5}^{68} - \)\(21\!\cdots\!78\)\( T_{5}^{67} - \)\(13\!\cdots\!48\)\( T_{5}^{66} - \)\(11\!\cdots\!14\)\( T_{5}^{65} + \)\(39\!\cdots\!89\)\( T_{5}^{64} + \)\(14\!\cdots\!31\)\( T_{5}^{63} + \)\(21\!\cdots\!23\)\( T_{5}^{62} + \)\(13\!\cdots\!81\)\( T_{5}^{61} - \)\(66\!\cdots\!15\)\( T_{5}^{60} - \)\(13\!\cdots\!66\)\( T_{5}^{59} + \)\(44\!\cdots\!27\)\( T_{5}^{58} + \)\(17\!\cdots\!36\)\( T_{5}^{57} + \)\(29\!\cdots\!64\)\( T_{5}^{56} + \)\(27\!\cdots\!01\)\( T_{5}^{55} + \)\(15\!\cdots\!90\)\( T_{5}^{54} + \)\(13\!\cdots\!24\)\( T_{5}^{53} + \)\(54\!\cdots\!39\)\( T_{5}^{52} + \)\(13\!\cdots\!57\)\( T_{5}^{51} + \)\(20\!\cdots\!92\)\( T_{5}^{50} + \)\(21\!\cdots\!39\)\( T_{5}^{49} + \)\(16\!\cdots\!07\)\( T_{5}^{48} + \)\(11\!\cdots\!27\)\( T_{5}^{47} + \)\(83\!\cdots\!65\)\( T_{5}^{46} + \)\(87\!\cdots\!98\)\( T_{5}^{45} + \)\(87\!\cdots\!00\)\( T_{5}^{44} + \)\(58\!\cdots\!57\)\( T_{5}^{43} + \)\(21\!\cdots\!52\)\( T_{5}^{42} + \)\(31\!\cdots\!51\)\( T_{5}^{41} + \)\(31\!\cdots\!66\)\( T_{5}^{40} + \)\(48\!\cdots\!42\)\( T_{5}^{39} + \)\(18\!\cdots\!16\)\( T_{5}^{38} - \)\(13\!\cdots\!40\)\( T_{5}^{37} - \)\(95\!\cdots\!22\)\( T_{5}^{36} + \)\(86\!\cdots\!49\)\( T_{5}^{35} + \)\(12\!\cdots\!36\)\( T_{5}^{34} + \)\(51\!\cdots\!30\)\( T_{5}^{33} - \)\(47\!\cdots\!29\)\( T_{5}^{32} - \)\(10\!\cdots\!89\)\( T_{5}^{31} - \)\(22\!\cdots\!28\)\( T_{5}^{30} + \)\(98\!\cdots\!20\)\( T_{5}^{29} - \)\(11\!\cdots\!96\)\( T_{5}^{28} - \)\(11\!\cdots\!43\)\( T_{5}^{27} - \)\(19\!\cdots\!19\)\( T_{5}^{26} + \)\(24\!\cdots\!71\)\( T_{5}^{25} + \)\(18\!\cdots\!61\)\( T_{5}^{24} + \)\(79\!\cdots\!20\)\( T_{5}^{23} + \)\(37\!\cdots\!23\)\( T_{5}^{22} + \)\(15\!\cdots\!46\)\( T_{5}^{21} + \)\(25\!\cdots\!85\)\( T_{5}^{20} - \)\(48\!\cdots\!35\)\( T_{5}^{19} + \)\(33\!\cdots\!22\)\( T_{5}^{18} + \)\(68\!\cdots\!01\)\( T_{5}^{17} - \)\(34\!\cdots\!69\)\( T_{5}^{16} - \)\(33\!\cdots\!40\)\( T_{5}^{15} - \)\(83\!\cdots\!33\)\( T_{5}^{14} + \)\(13\!\cdots\!85\)\( T_{5}^{13} + \)\(95\!\cdots\!11\)\( T_{5}^{12} + \)\(12\!\cdots\!40\)\( T_{5}^{11} + \)\(80\!\cdots\!24\)\( T_{5}^{10} + \)\(59\!\cdots\!39\)\( T_{5}^{9} + \)\(53\!\cdots\!45\)\( T_{5}^{8} - \)\(28\!\cdots\!03\)\( T_{5}^{7} + \)\(85\!\cdots\!47\)\( T_{5}^{6} + \)\(55\!\cdots\!43\)\( T_{5}^{5} + \)\(32\!\cdots\!70\)\( T_{5}^{4} - \)\(10\!\cdots\!96\)\( T_{5}^{3} - \)\(80\!\cdots\!39\)\( T_{5}^{2} - \)\(76\!\cdots\!97\)\( T_{5} + \)\(10\!\cdots\!01\)\( \)">\(T_{5}^{160} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).