Properties

Label 690.2.q.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{4} - 16q^{5} - 2q^{6} + 42q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{4} - 16q^{5} - 2q^{6} + 42q^{9} - 12q^{11} - 12q^{14} - 16q^{16} - 8q^{18} + 16q^{20} + 62q^{21} + 4q^{23} + 2q^{24} - 16q^{25} + 42q^{27} - 2q^{30} - 4q^{31} + 16q^{33} + 2q^{36} + 72q^{38} - 124q^{39} + 44q^{41} + 44q^{43} + 12q^{44} - 2q^{45} + 4q^{46} + 70q^{49} - 2q^{51} - 52q^{53} + 92q^{54} + 10q^{55} - 54q^{56} - 38q^{57} - 36q^{58} - 44q^{61} - 220q^{63} + 16q^{64} - 34q^{66} - 44q^{67} + 22q^{69} - 12q^{70} - 36q^{72} - 28q^{73} - 24q^{74} - 88q^{77} - 54q^{78} - 44q^{79} - 16q^{80} - 66q^{81} - 28q^{82} + 4q^{83} - 18q^{84} + 158q^{86} - 64q^{87} + 80q^{89} - 8q^{90} - 4q^{92} + 4q^{93} + 24q^{94} - 2q^{96} - 88q^{98} + 190q^{99} + O(q^{100}) \)

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 −1.73087 + 0.0639912i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.989611 + 1.42150i 0.918508 0.795892i 0.989821 0.142315i 2.99181 0.221521i 0.755750 + 0.654861i
11.2 −0.540641 0.841254i −1.61107 + 0.635963i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.40602 + 1.01149i −3.41511 + 2.95921i 0.989821 0.142315i 2.19110 2.04916i 0.755750 + 0.654861i
11.3 −0.540641 0.841254i −1.10251 1.33584i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.527715 + 1.64970i −3.34609 + 2.89940i 0.989821 0.142315i −0.568927 + 2.94556i 0.755750 + 0.654861i
11.4 −0.540641 0.841254i −0.717428 + 1.57648i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.71409 0.248772i 3.34933 2.90221i 0.989821 0.142315i −1.97059 2.26203i 0.755750 + 0.654861i
11.5 −0.540641 0.841254i 0.0741266 1.73046i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.49583 + 0.873200i 1.57600 1.36561i 0.989821 0.142315i −2.98901 0.256547i 0.755750 + 0.654861i
11.6 −0.540641 0.841254i 1.20125 + 1.24780i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.400274 1.68516i −1.03609 + 0.897775i 0.989821 0.142315i −0.114014 + 2.99783i 0.755750 + 0.654861i
11.7 −0.540641 0.841254i 1.47579 0.906664i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.56061 0.751335i −1.93677 + 1.67822i 0.989821 0.142315i 1.35592 2.67609i 0.755750 + 0.654861i
11.8 −0.540641 0.841254i 1.50109 + 0.864143i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.0845842 1.72998i 0.920755 0.797839i 0.989821 0.142315i 1.50651 + 2.59431i 0.755750 + 0.654861i
11.9 0.540641 + 0.841254i −1.62851 0.589888i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.384191 1.68890i 1.12640 0.976034i −0.989821 + 0.142315i 2.30406 + 1.92127i −0.755750 0.654861i
11.10 0.540641 + 0.841254i −1.25644 + 1.19221i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.68223 0.412423i 3.70222 3.20799i −0.989821 + 0.142315i 0.157263 2.99588i −0.755750 0.654861i
11.11 0.540641 + 0.841254i −0.549229 1.64266i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.08496 1.35013i −0.315364 + 0.273265i −0.989821 + 0.142315i −2.39669 + 1.80440i −0.755750 0.654861i
11.12 0.540641 + 0.841254i −0.162462 + 1.72441i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.53850 + 0.795618i 0.461235 0.399662i −0.989821 + 0.142315i −2.94721 0.560302i −0.755750 0.654861i
11.13 0.540641 + 0.841254i 0.559698 1.63913i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.68152 0.415331i −2.94981 + 2.55602i −0.989821 + 0.142315i −2.37348 1.83483i −0.755750 0.654861i
11.14 0.540641 + 0.841254i 0.843944 + 1.51253i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.816154 + 1.52771i −1.57741 + 1.36683i −0.989821 + 0.142315i −1.57552 + 2.55299i −0.755750 0.654861i
11.15 0.540641 + 0.841254i 1.37259 1.05641i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.63079 + 0.583554i 2.43170 2.10708i −0.989821 + 0.142315i 0.767988 2.90003i −0.755750 0.654861i
11.16 0.540641 + 0.841254i 1.73004 + 0.0835168i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.865069 + 1.50055i 0.0904807 0.0784019i −0.989821 + 0.142315i 2.98605 + 0.288974i −0.755750 0.654861i
191.1 −0.755750 0.654861i −1.72862 + 0.108947i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.37775 + 1.04967i −0.501114 1.70664i 0.540641 0.841254i 2.97626 0.376654i 0.281733 0.959493i
191.2 −0.755750 0.654861i −1.23307 1.21636i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.135347 + 1.72675i 0.0399458 + 0.136043i 0.540641 0.841254i 0.0409354 + 2.99972i 0.281733 0.959493i
191.3 −0.755750 0.654861i −0.906618 + 1.47582i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.65163 0.521642i −0.302591 1.03053i 0.540641 0.841254i −1.35609 2.67601i 0.281733 0.959493i
191.4 −0.755750 0.654861i 0.359391 1.69435i 0.142315 + 0.989821i 0.415415 + 0.909632i −1.38118 + 1.04516i 0.816450 + 2.78057i 0.540641 0.841254i −2.74168 1.21787i 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.a 160
3.b odd 2 1 690.2.q.b yes 160
23.d odd 22 1 690.2.q.b yes 160
69.g even 22 1 inner 690.2.q.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.q.a 160 1.a even 1 1 trivial
690.2.q.a 160 69.g even 22 1 inner
690.2.q.b yes 160 3.b odd 2 1
690.2.q.b yes 160 23.d odd 22 1

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 \(S_{2}^{\mathrm{new}}(690, [\chi])\).