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 $4$

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.

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

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