Properties

Label 690.2.w.b
Level $690$
Weight $2$
Character orbit 690.w
Analytic conductor $5.510$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.w (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{6} - 44q^{10} + 24q^{16} - 44q^{21} + 96q^{23} + 16q^{25} + 16q^{26} + 44q^{28} - 16q^{31} + 44q^{33} + 16q^{35} - 24q^{36} + 44q^{37} - 88q^{43} + 8q^{46} + 96q^{47} - 24q^{50} - 24q^{55} + 44q^{57} - 16q^{58} + 88q^{61} + 56q^{62} - 88q^{65} + 132q^{67} - 56q^{70} + 16q^{71} + 48q^{73} + 24q^{81} - 24q^{82} + 44q^{85} - 16q^{87} + 44q^{88} - 124q^{92} + 32q^{93} + 20q^{95} - 24q^{96} - 56q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.95274 + 1.08941i 0.142315 0.989821i 1.32241 2.42181i −0.977147 + 0.212565i −0.909632 0.415415i 1.87005 1.22594i
7.2 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.89440 1.18796i 0.142315 0.989821i 0.244741 0.448210i −0.977147 + 0.212565i −0.909632 0.415415i 1.97432 + 1.04979i
7.3 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −0.619349 2.14858i 0.142315 0.989821i −1.99507 + 3.65370i −0.977147 + 0.212565i −0.909632 0.415415i 0.771049 + 2.09892i
7.4 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.62532 + 1.53569i 0.142315 0.989821i 1.83156 3.35425i −0.977147 + 0.212565i −0.909632 0.415415i −1.73074 1.41582i
7.5 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.87213 1.22275i 0.142315 0.989821i −0.241465 + 0.442209i −0.977147 + 0.212565i −0.909632 0.415415i −1.78013 + 1.35319i
7.6 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.98312 + 1.03307i 0.142315 0.989821i −0.653533 + 1.19686i −0.977147 + 0.212565i −0.909632 0.415415i −2.05177 0.888964i
7.7 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.18918 + 0.455502i 0.142315 0.989821i 0.172948 0.316731i 0.977147 0.212565i −0.909632 0.415415i −2.15111 + 0.610516i
7.8 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −1.40926 1.73608i 0.142315 0.989821i 1.26240 2.31192i 0.977147 0.212565i −0.909632 0.415415i −1.52952 1.63113i
7.9 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.0285953 + 2.23589i 0.142315 0.989821i −1.81898 + 3.33121i 0.977147 0.212565i −0.909632 0.415415i 0.188029 + 2.22815i
7.10 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.855176 + 2.06608i 0.142315 0.989821i 1.43032 2.61943i 0.977147 0.212565i −0.909632 0.415415i 1.00039 + 1.99981i
7.11 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.40256 1.74150i 0.142315 0.989821i 1.87882 3.44079i 0.977147 0.212565i −0.909632 0.415415i 1.27475 1.83712i
7.12 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.93237 1.12513i 0.142315 0.989821i −0.968092 + 1.77293i 0.977147 0.212565i −0.909632 0.415415i 1.84719 1.26012i
37.1 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −2.21203 0.326962i −0.841254 0.540641i 0.672927 + 1.80419i −0.0713392 0.997452i 0.989821 + 0.142315i 1.78476 + 1.34708i
37.2 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −1.79792 1.32947i −0.841254 0.540641i −1.02771 2.75539i −0.0713392 0.997452i 0.989821 + 0.142315i 0.940853 + 2.02850i
37.3 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 0.217321 2.22548i −0.841254 0.540641i 0.154967 + 0.415484i −0.0713392 0.997452i 0.989821 + 0.142315i −1.25730 + 1.84911i
37.4 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 1.06201 + 1.96778i −0.841254 0.540641i −0.702461 1.88337i −0.0713392 0.997452i 0.989821 + 0.142315i 0.0109547 2.23604i
37.5 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.06085 + 0.867694i −0.841254 0.540641i −1.60657 4.30737i −0.0713392 0.997452i 0.989821 + 0.142315i −1.39292 1.74922i
37.6 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.18586 0.471182i −0.841254 0.540641i 1.33355 + 3.57540i −0.0713392 0.997452i 0.989821 + 0.142315i −2.14430 0.634025i
37.7 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −2.09572 0.779713i −0.841254 0.540641i −0.448960 1.20371i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.46569 1.68871i
37.8 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −1.84527 1.26292i −0.841254 0.540641i 1.33335 + 3.57484i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.01431 1.99278i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.w.b 240
5.c odd 4 1 inner 690.2.w.b 240
23.d odd 22 1 inner 690.2.w.b 240
115.l even 44 1 inner 690.2.w.b 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.w.b 240 1.a even 1 1 trivial
690.2.w.b 240 5.c odd 4 1 inner
690.2.w.b 240 23.d odd 22 1 inner
690.2.w.b 240 115.l even 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!56\)\( T_{7}^{221} - \)\(71\!\cdots\!54\)\( T_{7}^{220} + \)\(81\!\cdots\!76\)\( T_{7}^{219} + \)\(30\!\cdots\!44\)\( T_{7}^{218} - \)\(57\!\cdots\!36\)\( T_{7}^{217} + \)\(41\!\cdots\!81\)\( T_{7}^{216} - \)\(69\!\cdots\!40\)\( T_{7}^{215} - \)\(23\!\cdots\!28\)\( T_{7}^{214} + \)\(40\!\cdots\!92\)\( T_{7}^{213} - \)\(24\!\cdots\!56\)\( T_{7}^{212} + \)\(51\!\cdots\!04\)\( T_{7}^{211} + \)\(16\!\cdots\!88\)\( T_{7}^{210} - \)\(25\!\cdots\!44\)\( T_{7}^{209} + \)\(13\!\cdots\!63\)\( T_{7}^{208} - \)\(29\!\cdots\!40\)\( T_{7}^{207} - \)\(79\!\cdots\!16\)\( T_{7}^{206} + \)\(11\!\cdots\!44\)\( T_{7}^{205} - \)\(57\!\cdots\!46\)\( T_{7}^{204} + \)\(10\!\cdots\!76\)\( T_{7}^{203} + \)\(34\!\cdots\!48\)\( T_{7}^{202} - \)\(43\!\cdots\!28\)\( T_{7}^{201} + \)\(21\!\cdots\!20\)\( T_{7}^{200} - \)\(39\!\cdots\!64\)\( T_{7}^{199} - \)\(12\!\cdots\!44\)\( T_{7}^{198} + \)\(14\!\cdots\!16\)\( T_{7}^{197} - \)\(64\!\cdots\!90\)\( T_{7}^{196} + \)\(98\!\cdots\!12\)\( T_{7}^{195} + \)\(45\!\cdots\!28\)\( T_{7}^{194} - \)\(45\!\cdots\!44\)\( T_{7}^{193} + \)\(19\!\cdots\!79\)\( T_{7}^{192} - \)\(25\!\cdots\!00\)\( T_{7}^{191} - \)\(13\!\cdots\!32\)\( T_{7}^{190} + \)\(11\!\cdots\!24\)\( T_{7}^{189} - \)\(43\!\cdots\!84\)\( T_{7}^{188} + \)\(28\!\cdots\!96\)\( T_{7}^{187} + \)\(38\!\cdots\!12\)\( T_{7}^{186} - \)\(26\!\cdots\!72\)\( T_{7}^{185} + \)\(91\!\cdots\!47\)\( T_{7}^{184} - \)\(37\!\cdots\!08\)\( T_{7}^{183} - \)\(79\!\cdots\!16\)\( T_{7}^{182} + \)\(52\!\cdots\!92\)\( T_{7}^{181} - \)\(17\!\cdots\!28\)\( T_{7}^{180} - \)\(67\!\cdots\!08\)\( T_{7}^{179} + \)\(17\!\cdots\!68\)\( T_{7}^{178} - \)\(90\!\cdots\!52\)\( T_{7}^{177} + \)\(24\!\cdots\!58\)\( T_{7}^{176} + \)\(31\!\cdots\!76\)\( T_{7}^{175} - \)\(26\!\cdots\!20\)\( T_{7}^{174} + \)\(14\!\cdots\!44\)\( T_{7}^{173} - \)\(37\!\cdots\!31\)\( T_{7}^{172} - \)\(77\!\cdots\!32\)\( T_{7}^{171} + \)\(46\!\cdots\!36\)\( T_{7}^{170} - \)\(16\!\cdots\!52\)\( T_{7}^{169} + \)\(34\!\cdots\!93\)\( T_{7}^{168} + \)\(17\!\cdots\!36\)\( T_{7}^{167} - \)\(41\!\cdots\!04\)\( T_{7}^{166} + \)\(21\!\cdots\!52\)\( T_{7}^{165} - \)\(15\!\cdots\!17\)\( T_{7}^{164} - \)\(24\!\cdots\!48\)\( T_{7}^{163} + \)\(46\!\cdots\!08\)\( T_{7}^{162} - \)\(37\!\cdots\!52\)\( T_{7}^{161} + \)\(22\!\cdots\!91\)\( T_{7}^{160} + \)\(27\!\cdots\!04\)\( T_{7}^{159} + \)\(15\!\cdots\!20\)\( T_{7}^{158} + \)\(23\!\cdots\!24\)\( T_{7}^{157} + \)\(76\!\cdots\!80\)\( T_{7}^{156} - \)\(15\!\cdots\!28\)\( T_{7}^{155} + \)\(47\!\cdots\!32\)\( T_{7}^{154} + \)\(84\!\cdots\!84\)\( T_{7}^{153} + \)\(21\!\cdots\!66\)\( T_{7}^{152} + \)\(12\!\cdots\!64\)\( T_{7}^{151} + \)\(32\!\cdots\!16\)\( T_{7}^{150} + \)\(14\!\cdots\!92\)\( T_{7}^{149} + \)\(72\!\cdots\!24\)\( T_{7}^{148} + \)\(32\!\cdots\!16\)\( T_{7}^{147} - \)\(11\!\cdots\!20\)\( T_{7}^{146} + \)\(31\!\cdots\!72\)\( T_{7}^{145} + \)\(31\!\cdots\!15\)\( T_{7}^{144} + \)\(76\!\cdots\!48\)\( T_{7}^{143} + \)\(17\!\cdots\!16\)\( T_{7}^{142} - \)\(79\!\cdots\!76\)\( T_{7}^{141} - \)\(35\!\cdots\!78\)\( T_{7}^{140} - \)\(15\!\cdots\!88\)\( T_{7}^{139} - \)\(54\!\cdots\!60\)\( T_{7}^{138} - \)\(12\!\cdots\!12\)\( T_{7}^{137} + \)\(74\!\cdots\!03\)\( T_{7}^{136} + \)\(13\!\cdots\!96\)\( T_{7}^{135} + \)\(62\!\cdots\!00\)\( T_{7}^{134} + \)\(17\!\cdots\!92\)\( T_{7}^{133} + \)\(32\!\cdots\!65\)\( T_{7}^{132} + \)\(23\!\cdots\!28\)\( T_{7}^{131} - \)\(11\!\cdots\!84\)\( T_{7}^{130} - \)\(75\!\cdots\!08\)\( T_{7}^{129} - \)\(27\!\cdots\!45\)\( T_{7}^{128} - \)\(66\!\cdots\!12\)\( T_{7}^{127} - \)\(88\!\cdots\!04\)\( T_{7}^{126} + \)\(30\!\cdots\!24\)\( T_{7}^{125} + \)\(43\!\cdots\!04\)\( T_{7}^{124} + \)\(11\!\cdots\!16\)\( T_{7}^{123} + \)\(24\!\cdots\!80\)\( T_{7}^{122} + \)\(58\!\cdots\!52\)\( T_{7}^{121} + \)\(13\!\cdots\!82\)\( T_{7}^{120} + \)\(18\!\cdots\!48\)\( T_{7}^{119} + \)\(19\!\cdots\!36\)\( T_{7}^{118} + \)\(11\!\cdots\!84\)\( T_{7}^{117} + \)\(59\!\cdots\!30\)\( T_{7}^{116} + \)\(13\!\cdots\!28\)\( T_{7}^{115} + \)\(92\!\cdots\!44\)\( T_{7}^{114} - \)\(29\!\cdots\!84\)\( T_{7}^{113} - \)\(40\!\cdots\!18\)\( T_{7}^{112} + \)\(12\!\cdots\!40\)\( T_{7}^{111} + \)\(40\!\cdots\!52\)\( T_{7}^{110} + \)\(11\!\cdots\!28\)\( T_{7}^{109} - \)\(85\!\cdots\!42\)\( T_{7}^{108} - \)\(99\!\cdots\!20\)\( T_{7}^{107} + \)\(60\!\cdots\!68\)\( T_{7}^{106} + \)\(86\!\cdots\!48\)\( T_{7}^{105} - \)\(12\!\cdots\!75\)\( T_{7}^{104} - \)\(60\!\cdots\!00\)\( T_{7}^{103} - \)\(54\!\cdots\!64\)\( T_{7}^{102} + \)\(89\!\cdots\!76\)\( T_{7}^{101} + \)\(39\!\cdots\!11\)\( T_{7}^{100} + \)\(60\!\cdots\!40\)\( T_{7}^{99} + \)\(44\!\cdots\!56\)\( T_{7}^{98} - \)\(12\!\cdots\!76\)\( T_{7}^{97} - \)\(58\!\cdots\!43\)\( T_{7}^{96} - \)\(11\!\cdots\!76\)\( T_{7}^{95} - \)\(31\!\cdots\!32\)\( T_{7}^{94} + \)\(35\!\cdots\!28\)\( T_{7}^{93} + \)\(85\!\cdots\!54\)\( T_{7}^{92} + \)\(47\!\cdots\!80\)\( T_{7}^{91} - \)\(14\!\cdots\!16\)\( T_{7}^{90} - \)\(29\!\cdots\!72\)\( T_{7}^{89} + \)\(91\!\cdots\!80\)\( T_{7}^{88} + \)\(93\!\cdots\!68\)\( T_{7}^{87} + \)\(60\!\cdots\!84\)\( T_{7}^{86} - \)\(26\!\cdots\!16\)\( T_{7}^{85} - \)\(52\!\cdots\!43\)\( T_{7}^{84} + \)\(30\!\cdots\!48\)\( T_{7}^{83} + \)\(22\!\cdots\!72\)\( T_{7}^{82} + \)\(15\!\cdots\!88\)\( T_{7}^{81} - \)\(79\!\cdots\!93\)\( T_{7}^{80} - \)\(22\!\cdots\!52\)\( T_{7}^{79} - \)\(15\!\cdots\!72\)\( T_{7}^{78} + \)\(49\!\cdots\!52\)\( T_{7}^{77} + \)\(16\!\cdots\!30\)\( T_{7}^{76} + \)\(18\!\cdots\!96\)\( T_{7}^{75} - \)\(53\!\cdots\!76\)\( T_{7}^{74} - \)\(59\!\cdots\!60\)\( T_{7}^{73} - \)\(10\!\cdots\!29\)\( T_{7}^{72} - \)\(62\!\cdots\!84\)\( T_{7}^{71} + \)\(10\!\cdots\!28\)\( T_{7}^{70} + \)\(30\!\cdots\!12\)\( T_{7}^{69} + \)\(31\!\cdots\!00\)\( T_{7}^{68} - \)\(37\!\cdots\!20\)\( T_{7}^{67} - \)\(56\!\cdots\!44\)\( T_{7}^{66} - \)\(76\!\cdots\!24\)\( T_{7}^{65} - \)\(21\!\cdots\!82\)\( T_{7}^{64} + \)\(87\!\cdots\!28\)\( T_{7}^{63} + \)\(17\!\cdots\!40\)\( T_{7}^{62} + \)\(15\!\cdots\!84\)\( T_{7}^{61} + \)\(53\!\cdots\!53\)\( T_{7}^{60} - \)\(55\!\cdots\!40\)\( T_{7}^{59} - \)\(96\!\cdots\!32\)\( T_{7}^{58} - \)\(67\!\cdots\!88\)\( T_{7}^{57} - \)\(20\!\cdots\!36\)\( T_{7}^{56} + \)\(78\!\cdots\!16\)\( T_{7}^{55} - \)\(50\!\cdots\!44\)\( T_{7}^{54} - \)\(11\!\cdots\!20\)\( T_{7}^{53} - \)\(41\!\cdots\!35\)\( T_{7}^{52} + \)\(77\!\cdots\!88\)\( T_{7}^{51} + \)\(11\!\cdots\!16\)\( T_{7}^{50} + \)\(70\!\cdots\!20\)\( T_{7}^{49} + \)\(12\!\cdots\!05\)\( T_{7}^{48} + \)\(17\!\cdots\!32\)\( T_{7}^{47} + \)\(18\!\cdots\!80\)\( T_{7}^{46} + \)\(30\!\cdots\!56\)\( T_{7}^{45} + \)\(10\!\cdots\!58\)\( T_{7}^{44} - \)\(20\!\cdots\!76\)\( T_{7}^{43} - \)\(19\!\cdots\!36\)\( T_{7}^{42} - \)\(18\!\cdots\!32\)\( T_{7}^{41} - \)\(54\!\cdots\!35\)\( T_{7}^{40} + \)\(65\!\cdots\!76\)\( T_{7}^{39} + \)\(22\!\cdots\!20\)\( T_{7}^{38} + \)\(37\!\cdots\!00\)\( T_{7}^{37} + \)\(18\!\cdots\!88\)\( T_{7}^{36} - \)\(45\!\cdots\!68\)\( T_{7}^{35} + \)\(40\!\cdots\!76\)\( T_{7}^{34} - \)\(17\!\cdots\!20\)\( T_{7}^{33} - \)\(16\!\cdots\!76\)\( T_{7}^{32} + \)\(99\!\cdots\!44\)\( T_{7}^{31} - \)\(71\!\cdots\!16\)\( T_{7}^{30} + \)\(43\!\cdots\!64\)\( T_{7}^{29} + \)\(15\!\cdots\!72\)\( T_{7}^{28} - \)\(21\!\cdots\!64\)\( T_{7}^{27} - \)\(71\!\cdots\!60\)\( T_{7}^{26} - \)\(73\!\cdots\!64\)\( T_{7}^{25} - \)\(54\!\cdots\!48\)\( T_{7}^{24} + \)\(17\!\cdots\!44\)\( T_{7}^{23} - \)\(11\!\cdots\!96\)\( T_{7}^{22} + \)\(17\!\cdots\!04\)\( T_{7}^{21} + \)\(58\!\cdots\!48\)\( T_{7}^{20} - \)\(20\!\cdots\!96\)\( T_{7}^{19} + \)\(25\!\cdots\!24\)\( T_{7}^{18} + \)\(21\!\cdots\!40\)\( T_{7}^{17} - \)\(19\!\cdots\!96\)\( T_{7}^{16} - \)\(31\!\cdots\!64\)\( T_{7}^{15} + \)\(16\!\cdots\!76\)\( T_{7}^{14} - \)\(68\!\cdots\!92\)\( T_{7}^{13} + \)\(11\!\cdots\!88\)\( T_{7}^{12} + \)\(98\!\cdots\!32\)\( T_{7}^{11} - \)\(53\!\cdots\!56\)\( T_{7}^{10} + \)\(44\!\cdots\!56\)\( T_{7}^{9} + \)\(16\!\cdots\!08\)\( T_{7}^{8} - \)\(18\!\cdots\!00\)\( T_{7}^{7} + \)\(90\!\cdots\!68\)\( T_{7}^{6} - \)\(35\!\cdots\!44\)\( T_{7}^{5} + \)\(11\!\cdots\!48\)\( T_{7}^{4} - \)\(27\!\cdots\!16\)\( T_{7}^{3} + \)\(51\!\cdots\!28\)\( T_{7}^{2} - \)\(75\!\cdots\!76\)\( T_{7} + \)\(55\!\cdots\!96\)\( \)">\(T_{7}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).