Properties

Label 690.2.q.b
Level $690$
Weight $2$
Character orbit 690.q
Analytic conductor $5.510$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.q (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{4} + 16q^{5} - 2q^{6} - 46q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{4} + 16q^{5} - 2q^{6} - 46q^{9} + 12q^{11} + 12q^{14} - 16q^{16} - 8q^{18} - 16q^{20} + 70q^{21} - 4q^{23} + 2q^{24} - 16q^{25} + 42q^{27} + 2q^{30} - 4q^{31} - 16q^{33} + 2q^{36} - 72q^{38} + 140q^{39} - 44q^{41} + 44q^{43} - 12q^{44} + 2q^{45} + 4q^{46} + 70q^{49} + 2q^{51} + 52q^{53} - 62q^{54} + 10q^{55} + 54q^{56} - 94q^{57} - 36q^{58} - 44q^{61} + 16q^{64} - 54q^{66} - 44q^{67} - 30q^{69} - 12q^{70} - 36q^{72} - 28q^{73} + 24q^{74} + 88q^{77} - 54q^{78} - 44q^{79} + 16q^{80} - 66q^{81} - 28q^{82} - 4q^{83} - 4q^{84} - 158q^{86} + 156q^{87} - 80q^{89} + 8q^{90} + 4q^{92} + 4q^{93} + 24q^{94} - 2q^{96} + 88q^{98} - 58q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.540641 0.841254i −1.68349 0.407274i −0.415415 + 0.909632i 0.959493 0.281733i 0.567541 + 1.63643i 0.0904807 0.0784019i 0.989821 0.142315i 2.66826 + 1.37128i −0.755750 0.654861i
11.2 −0.540641 0.841254i −1.23589 + 1.21350i −0.415415 + 0.909632i 0.959493 0.281733i 1.68903 + 0.383629i −1.57741 + 1.36683i 0.989821 0.142315i 0.0548407 2.99950i −0.755750 0.654861i
11.3 −0.540641 0.841254i −1.01936 1.40032i −0.415415 + 0.909632i 0.959493 0.281733i −0.626917 + 1.61461i 2.43170 2.10708i 0.989821 0.142315i −0.921804 + 2.85487i −0.755750 0.654861i
11.4 −0.540641 0.841254i −0.329943 + 1.70033i −0.415415 + 0.909632i 0.959493 0.281733i 1.60879 0.641705i 0.461235 0.399662i 0.989821 0.142315i −2.78228 1.12203i −0.755750 0.654861i
11.5 −0.540641 0.841254i −0.0752308 1.73042i −0.415415 + 0.909632i 0.959493 0.281733i −1.41505 + 0.998822i −2.94981 + 2.55602i 0.989821 0.142315i −2.98868 + 0.260361i −0.755750 0.654861i
11.6 −0.540641 0.841254i 0.869657 + 1.49790i −0.415415 + 0.909632i 0.959493 0.281733i 0.789940 1.54143i 3.70222 3.20799i 0.989821 0.142315i −1.48739 + 2.60531i −0.755750 0.654861i
11.7 −0.540641 0.841254i 0.989774 1.42139i −0.415415 + 0.909632i 0.959493 0.281733i −1.73086 0.0641895i −0.315364 + 0.273265i 0.989821 0.142315i −1.04070 2.81371i −0.755750 0.654861i
11.8 −0.540641 0.841254i 1.72873 0.107190i −0.415415 + 0.909632i 0.959493 0.281733i −1.02480 1.39635i 1.12640 0.976034i 0.989821 0.142315i 2.97702 0.370607i −0.755750 0.654861i
11.9 0.540641 + 0.841254i −1.68374 + 0.406235i −0.415415 + 0.909632i 0.959493 0.281733i −1.25204 1.19682i 0.920755 0.797839i −0.989821 + 0.142315i 2.66995 1.36799i 0.755750 + 0.654861i
11.10 0.540641 + 0.841254i −1.50413 + 0.858826i −0.415415 + 0.909632i 0.959493 0.281733i −1.53569 0.801041i −1.03609 + 0.897775i −0.989821 + 0.142315i 1.52484 2.58358i 0.755750 + 0.654861i
11.11 0.540641 + 0.841254i −1.16057 1.28572i −0.415415 + 0.909632i 0.959493 0.281733i 0.454160 1.67145i −1.93677 + 1.67822i −0.989821 + 0.142315i −0.306134 + 2.98434i 0.755750 + 0.654861i
11.12 0.540641 + 0.841254i 0.244221 + 1.71475i −0.415415 + 0.909632i 0.959493 0.281733i −1.31050 + 1.13251i 3.34933 2.90221i −0.989821 + 0.142315i −2.88071 + 0.837554i 0.755750 + 0.654861i
11.13 0.540641 + 0.841254i 0.416404 1.68125i −0.415415 + 0.909632i 0.959493 0.281733i 1.63948 0.558652i 1.57600 1.36561i −0.989821 + 0.142315i −2.65322 1.40016i 0.755750 + 0.654861i
11.14 0.540641 + 0.841254i 1.36664 + 1.06409i −0.415415 + 0.909632i 0.959493 0.281733i −0.156311 + 1.72498i −3.41511 + 2.95921i −0.989821 + 0.142315i 0.735409 + 2.90847i 0.755750 + 0.654861i
11.15 0.540641 + 0.841254i 1.43420 0.971113i −0.415415 + 0.909632i 0.959493 0.281733i 1.59234 + 0.681505i −3.34609 + 2.89940i −0.989821 + 0.142315i 1.11388 2.78555i 0.755750 + 0.654861i
11.16 0.540641 + 0.841254i 1.64273 + 0.549041i −0.415415 + 0.909632i 0.959493 0.281733i 0.426243 + 1.67878i 0.918508 0.795892i −0.989821 + 0.142315i 2.39711 + 1.80385i 0.755750 + 0.654861i
191.1 −0.755750 0.654861i −1.58079 0.707884i 0.142315 + 0.989821i −0.415415 0.909632i 0.731117 + 1.57018i 0.325415 + 1.10826i 0.540641 0.841254i 1.99780 + 2.23803i −0.281733 + 0.959493i
191.2 −0.755750 0.654861i −1.30022 + 1.14430i 0.142315 + 0.989821i −0.415415 0.909632i 1.73200 0.0133460i 0.519422 + 1.76899i 0.540641 0.841254i 0.381134 2.97569i −0.281733 + 0.959493i
191.3 −0.755750 0.654861i −0.982602 + 1.42636i 0.142315 + 0.989821i −0.415415 0.909632i 1.67667 0.434500i −0.819457 2.79081i 0.540641 0.841254i −1.06898 2.80308i −0.281733 + 0.959493i
191.4 −0.755750 0.654861i −0.340196 1.69831i 0.142315 + 0.989821i −0.415415 0.909632i −0.855055 + 1.50628i 0.680617 + 2.31797i 0.540641 0.841254i −2.76853 + 1.15552i −0.281733 + 0.959493i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.q.b yes 160
3.b odd 2 1 690.2.q.a 160
23.d odd 22 1 690.2.q.a 160
69.g even 22 1 inner 690.2.q.b yes 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.q.a 160 3.b odd 2 1
690.2.q.a 160 23.d odd 22 1
690.2.q.b yes 160 1.a even 1 1 trivial
690.2.q.b yes 160 69.g even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(12\!\cdots\!53\)\( T_{11}^{146} - \)\(58\!\cdots\!96\)\( T_{11}^{145} + \)\(46\!\cdots\!24\)\( T_{11}^{144} - \)\(22\!\cdots\!34\)\( T_{11}^{143} + \)\(17\!\cdots\!00\)\( T_{11}^{142} - \)\(93\!\cdots\!20\)\( T_{11}^{141} + \)\(67\!\cdots\!69\)\( T_{11}^{140} - \)\(32\!\cdots\!36\)\( T_{11}^{139} + \)\(22\!\cdots\!60\)\( T_{11}^{138} - \)\(10\!\cdots\!30\)\( T_{11}^{137} + \)\(70\!\cdots\!33\)\( T_{11}^{136} - \)\(34\!\cdots\!98\)\( T_{11}^{135} + \)\(22\!\cdots\!11\)\( T_{11}^{134} - \)\(11\!\cdots\!24\)\( T_{11}^{133} + \)\(73\!\cdots\!78\)\( T_{11}^{132} - \)\(37\!\cdots\!18\)\( T_{11}^{131} + \)\(22\!\cdots\!79\)\( T_{11}^{130} - \)\(11\!\cdots\!60\)\( T_{11}^{129} + \)\(67\!\cdots\!11\)\( T_{11}^{128} - \)\(34\!\cdots\!42\)\( T_{11}^{127} + \)\(19\!\cdots\!04\)\( T_{11}^{126} - \)\(94\!\cdots\!26\)\( T_{11}^{125} + \)\(51\!\cdots\!59\)\( T_{11}^{124} - \)\(25\!\cdots\!18\)\( T_{11}^{123} + \)\(13\!\cdots\!68\)\( T_{11}^{122} - \)\(63\!\cdots\!32\)\( T_{11}^{121} + \)\(32\!\cdots\!65\)\( T_{11}^{120} - \)\(15\!\cdots\!40\)\( T_{11}^{119} + \)\(76\!\cdots\!90\)\( T_{11}^{118} - \)\(36\!\cdots\!26\)\( T_{11}^{117} + \)\(18\!\cdots\!88\)\( T_{11}^{116} - \)\(86\!\cdots\!86\)\( T_{11}^{115} + \)\(42\!\cdots\!61\)\( T_{11}^{114} - \)\(20\!\cdots\!76\)\( T_{11}^{113} + \)\(96\!\cdots\!49\)\( T_{11}^{112} - \)\(43\!\cdots\!46\)\( T_{11}^{111} + \)\(19\!\cdots\!69\)\( T_{11}^{110} - \)\(85\!\cdots\!30\)\( T_{11}^{109} + \)\(36\!\cdots\!35\)\( T_{11}^{108} - \)\(14\!\cdots\!32\)\( T_{11}^{107} + \)\(60\!\cdots\!42\)\( T_{11}^{106} - \)\(23\!\cdots\!88\)\( T_{11}^{105} + \)\(90\!\cdots\!90\)\( T_{11}^{104} - \)\(33\!\cdots\!28\)\( T_{11}^{103} + \)\(12\!\cdots\!09\)\( T_{11}^{102} - \)\(41\!\cdots\!74\)\( T_{11}^{101} + \)\(14\!\cdots\!43\)\( T_{11}^{100} - \)\(47\!\cdots\!96\)\( T_{11}^{99} + \)\(15\!\cdots\!94\)\( T_{11}^{98} - \)\(47\!\cdots\!16\)\( T_{11}^{97} + \)\(14\!\cdots\!44\)\( T_{11}^{96} - \)\(42\!\cdots\!48\)\( T_{11}^{95} + \)\(13\!\cdots\!94\)\( T_{11}^{94} - \)\(34\!\cdots\!64\)\( T_{11}^{93} + \)\(10\!\cdots\!53\)\( T_{11}^{92} - \)\(26\!\cdots\!20\)\( T_{11}^{91} + \)\(80\!\cdots\!88\)\( T_{11}^{90} - \)\(18\!\cdots\!48\)\( T_{11}^{89} + \)\(61\!\cdots\!15\)\( T_{11}^{88} - \)\(13\!\cdots\!64\)\( T_{11}^{87} + \)\(51\!\cdots\!46\)\( T_{11}^{86} - \)\(11\!\cdots\!34\)\( T_{11}^{85} + \)\(44\!\cdots\!10\)\( T_{11}^{84} - \)\(94\!\cdots\!30\)\( T_{11}^{83} + \)\(38\!\cdots\!78\)\( T_{11}^{82} - \)\(70\!\cdots\!78\)\( T_{11}^{81} + \)\(26\!\cdots\!90\)\( T_{11}^{80} - \)\(31\!\cdots\!62\)\( T_{11}^{79} + \)\(13\!\cdots\!01\)\( T_{11}^{78} - \)\(24\!\cdots\!78\)\( T_{11}^{77} + \)\(72\!\cdots\!65\)\( T_{11}^{76} - \)\(20\!\cdots\!24\)\( T_{11}^{75} + \)\(75\!\cdots\!90\)\( T_{11}^{74} - \)\(91\!\cdots\!32\)\( T_{11}^{73} + \)\(65\!\cdots\!07\)\( T_{11}^{72} - \)\(65\!\cdots\!56\)\( T_{11}^{71} + \)\(32\!\cdots\!83\)\( T_{11}^{70} - \)\(94\!\cdots\!26\)\( T_{11}^{69} + \)\(11\!\cdots\!38\)\( T_{11}^{68} + \)\(28\!\cdots\!56\)\( T_{11}^{67} + \)\(64\!\cdots\!62\)\( T_{11}^{66} - \)\(54\!\cdots\!86\)\( T_{11}^{65} + \)\(44\!\cdots\!93\)\( T_{11}^{64} - \)\(51\!\cdots\!26\)\( T_{11}^{63} + \)\(20\!\cdots\!52\)\( T_{11}^{62} - \)\(17\!\cdots\!74\)\( T_{11}^{61} + \)\(71\!\cdots\!02\)\( T_{11}^{60} - \)\(93\!\cdots\!22\)\( T_{11}^{59} + \)\(36\!\cdots\!59\)\( T_{11}^{58} - \)\(67\!\cdots\!56\)\( T_{11}^{57} + \)\(19\!\cdots\!35\)\( T_{11}^{56} - \)\(28\!\cdots\!58\)\( T_{11}^{55} + \)\(62\!\cdots\!80\)\( T_{11}^{54} - \)\(79\!\cdots\!56\)\( T_{11}^{53} + \)\(12\!\cdots\!39\)\( T_{11}^{52} - \)\(15\!\cdots\!36\)\( T_{11}^{51} + \)\(49\!\cdots\!54\)\( T_{11}^{50} - \)\(54\!\cdots\!76\)\( T_{11}^{49} + \)\(17\!\cdots\!09\)\( T_{11}^{48} - \)\(31\!\cdots\!68\)\( T_{11}^{47} + \)\(45\!\cdots\!89\)\( T_{11}^{46} - \)\(70\!\cdots\!04\)\( T_{11}^{45} + \)\(17\!\cdots\!13\)\( T_{11}^{44} - \)\(15\!\cdots\!46\)\( T_{11}^{43} + \)\(38\!\cdots\!48\)\( T_{11}^{42} - \)\(77\!\cdots\!50\)\( T_{11}^{41} + \)\(98\!\cdots\!33\)\( T_{11}^{40} - \)\(15\!\cdots\!34\)\( T_{11}^{39} + \)\(28\!\cdots\!03\)\( T_{11}^{38} - \)\(23\!\cdots\!50\)\( T_{11}^{37} + \)\(29\!\cdots\!01\)\( T_{11}^{36} - \)\(37\!\cdots\!40\)\( T_{11}^{35} + \)\(21\!\cdots\!50\)\( T_{11}^{34} - \)\(11\!\cdots\!34\)\( T_{11}^{33} + \)\(27\!\cdots\!28\)\( T_{11}^{32} + \)\(11\!\cdots\!34\)\( T_{11}^{31} + \)\(12\!\cdots\!51\)\( T_{11}^{30} + \)\(49\!\cdots\!98\)\( T_{11}^{29} + \)\(50\!\cdots\!25\)\( T_{11}^{28} + \)\(25\!\cdots\!04\)\( T_{11}^{27} + \)\(11\!\cdots\!17\)\( T_{11}^{26} + \)\(46\!\cdots\!46\)\( T_{11}^{25} + \)\(58\!\cdots\!73\)\( T_{11}^{24} + \)\(26\!\cdots\!24\)\( T_{11}^{23} + \)\(13\!\cdots\!87\)\( T_{11}^{22} + \)\(64\!\cdots\!26\)\( T_{11}^{21} + \)\(29\!\cdots\!01\)\( T_{11}^{20} + \)\(44\!\cdots\!76\)\( T_{11}^{19} + \)\(17\!\cdots\!76\)\( T_{11}^{18} - \)\(12\!\cdots\!80\)\( T_{11}^{17} + \)\(63\!\cdots\!08\)\( T_{11}^{16} - \)\(49\!\cdots\!48\)\( T_{11}^{15} + \)\(10\!\cdots\!80\)\( T_{11}^{14} - \)\(26\!\cdots\!12\)\( T_{11}^{13} + \)\(11\!\cdots\!12\)\( T_{11}^{12} - \)\(11\!\cdots\!36\)\( T_{11}^{11} + \)\(51\!\cdots\!60\)\( T_{11}^{10} - \)\(84\!\cdots\!52\)\( T_{11}^{9} + \)\(73\!\cdots\!64\)\( T_{11}^{8} - \)\(73\!\cdots\!04\)\( T_{11}^{7} + \)\(91\!\cdots\!16\)\( T_{11}^{6} + \)\(39\!\cdots\!04\)\( T_{11}^{5} + \)\(68\!\cdots\!36\)\( T_{11}^{4} + \)\(41\!\cdots\!56\)\( T_{11}^{3} + \)\(91\!\cdots\!24\)\( T_{11}^{2} + \)\(98\!\cdots\!20\)\( T_{11} + \)\(12\!\cdots\!44\)\( \)">\(T_{11}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).