# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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