Properties

Label 690.2.w.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 240q - 24q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q - 24q^{6} + 44q^{10} - 16q^{13} + 24q^{16} + 44q^{21} + 72q^{23} + 16q^{25} + 44q^{28} - 16q^{31} - 44q^{33} - 24q^{36} + 44q^{37} + 88q^{43} - 8q^{46} + 48q^{47} + 8q^{50} - 16q^{52} + 56q^{55} + 44q^{57} + 16q^{58} + 88q^{61} + 8q^{62} + 88q^{65} - 132q^{67} + 56q^{70} - 64q^{71} + 16q^{73} - 32q^{75} - 16q^{77} - 16q^{78} + 24q^{81} - 24q^{82} + 92q^{85} - 16q^{87} - 44q^{88} + 116q^{92} - 80q^{93} + 20q^{95} + 24q^{96} - 88q^{98} + O(q^{100}) \)

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 −2.20400 + 0.377364i −0.142315 + 0.989821i 2.34384 4.29243i −0.977147 + 0.212565i −0.909632 0.415415i 2.17146 0.533633i
7.2 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.11721 0.719325i −0.142315 + 0.989821i −2.05801 + 3.76897i −0.977147 + 0.212565i −0.909632 0.415415i 2.16313 + 0.566452i
7.3 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.721297 + 2.11654i −0.142315 + 0.989821i −0.913900 + 1.67368i −0.977147 + 0.212565i −0.909632 0.415415i 0.568467 2.16260i
7.4 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.160701 2.23029i −0.142315 + 0.989821i 1.09028 1.99670i −0.977147 + 0.212565i −0.909632 0.415415i 0.319399 + 2.21314i
7.5 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.0140096 + 2.23602i −0.142315 + 0.989821i −0.151445 + 0.277351i −0.977147 + 0.212565i −0.909632 0.415415i −0.145542 2.23133i
7.6 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 2.23599 + 0.0191781i −0.142315 + 0.989821i 0.197874 0.362379i −0.977147 + 0.212565i −0.909632 0.415415i −2.23166 + 0.140384i
7.7 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −2.14628 0.627266i −0.142315 + 0.989821i −0.236933 + 0.433910i 0.977147 0.212565i −0.909632 0.415415i −2.18556 0.472554i
7.8 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.79544 + 1.33281i −0.142315 + 0.989821i 2.35788 4.31814i 0.977147 0.212565i −0.909632 0.415415i −1.69579 + 1.45750i
7.9 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.32705 1.79970i −0.142315 + 0.989821i 1.34666 2.46622i 0.977147 0.212565i −0.909632 0.415415i −1.45206 1.70045i
7.10 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −0.244725 + 2.22264i −0.142315 + 0.989821i −1.14160 + 2.09069i 0.977147 0.212565i −0.909632 0.415415i −0.0855405 + 2.23443i
7.11 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.24397 1.85810i −0.142315 + 0.989821i 0.294639 0.539591i 0.977147 0.212565i −0.909632 0.415415i 1.10824 1.94211i
7.12 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.68213 + 1.47324i −0.142315 + 0.989821i −0.663233 + 1.21462i 0.977147 0.212565i −0.909632 0.415415i 1.78294 + 1.34949i
37.1 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −2.15875 0.582912i 0.841254 + 0.540641i 0.0886059 + 0.237562i −0.0713392 0.997452i 0.989821 + 0.142315i 1.61533 + 1.54619i
37.2 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −1.27260 1.83861i 0.841254 + 0.540641i −1.45896 3.91162i −0.0713392 0.997452i 0.989821 + 0.142315i 0.235778 + 2.22360i
37.3 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −0.00639501 + 2.23606i 0.841254 + 0.540641i −1.30166 3.48989i −0.0713392 0.997452i 0.989821 + 0.142315i 1.07724 1.95948i
37.4 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 0.565910 + 2.16327i 0.841254 + 0.540641i 1.25538 + 3.36582i −0.0713392 0.997452i 0.989821 + 0.142315i 0.540059 2.16987i
37.5 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 2.11884 0.714499i 0.841254 + 0.540641i 0.271075 + 0.726780i −0.0713392 0.997452i 0.989821 + 0.142315i −2.20209 0.388352i
37.6 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 2.22914 0.175901i 0.841254 + 0.540641i −0.0297310 0.0797118i −0.0713392 0.997452i 0.989821 + 0.142315i −2.04077 0.913927i
37.7 0.877679 + 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −1.94930 + 1.09554i 0.841254 + 0.540641i 0.266002 + 0.713180i 0.0713392 + 0.997452i 0.989821 + 0.142315i −2.23590 + 0.0273353i
37.8 0.877679 + 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −0.516511 2.17560i 0.841254 + 0.540641i −1.41675 3.79846i 0.0713392 + 0.997452i 0.989821 + 0.142315i 0.589321 2.15701i
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.a 240
5.c odd 4 1 inner 690.2.w.a 240
23.d odd 22 1 inner 690.2.w.a 240
115.l even 44 1 inner 690.2.w.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.w.a 240 1.a even 1 1 trivial
690.2.w.a 240 5.c odd 4 1 inner
690.2.w.a 240 23.d odd 22 1 inner
690.2.w.a 240 115.l even 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!64\)\( T_{7}^{223} - \)\(69\!\cdots\!80\)\( T_{7}^{222} + \)\(35\!\cdots\!44\)\( T_{7}^{221} - \)\(82\!\cdots\!26\)\( T_{7}^{220} - \)\(78\!\cdots\!48\)\( T_{7}^{219} + \)\(31\!\cdots\!72\)\( T_{7}^{218} - \)\(18\!\cdots\!36\)\( T_{7}^{217} + \)\(16\!\cdots\!93\)\( T_{7}^{216} + \)\(37\!\cdots\!60\)\( T_{7}^{215} - \)\(13\!\cdots\!04\)\( T_{7}^{214} + \)\(82\!\cdots\!12\)\( T_{7}^{213} + \)\(22\!\cdots\!64\)\( T_{7}^{212} - \)\(14\!\cdots\!12\)\( T_{7}^{211} + \)\(57\!\cdots\!56\)\( T_{7}^{210} - \)\(31\!\cdots\!40\)\( T_{7}^{209} - \)\(57\!\cdots\!57\)\( T_{7}^{208} + \)\(58\!\cdots\!56\)\( T_{7}^{207} - \)\(19\!\cdots\!44\)\( T_{7}^{206} + \)\(11\!\cdots\!40\)\( T_{7}^{205} - \)\(55\!\cdots\!46\)\( T_{7}^{204} - \)\(23\!\cdots\!36\)\( T_{7}^{203} + \)\(56\!\cdots\!72\)\( T_{7}^{202} - \)\(37\!\cdots\!72\)\( T_{7}^{201} + \)\(34\!\cdots\!76\)\( T_{7}^{200} + \)\(85\!\cdots\!68\)\( T_{7}^{199} - \)\(14\!\cdots\!16\)\( T_{7}^{198} + \)\(13\!\cdots\!32\)\( T_{7}^{197} - \)\(11\!\cdots\!90\)\( T_{7}^{196} - \)\(31\!\cdots\!36\)\( T_{7}^{195} + \)\(27\!\cdots\!16\)\( T_{7}^{194} - \)\(51\!\cdots\!64\)\( T_{7}^{193} + \)\(14\!\cdots\!87\)\( T_{7}^{192} + \)\(11\!\cdots\!60\)\( T_{7}^{191} - \)\(14\!\cdots\!24\)\( T_{7}^{190} + \)\(21\!\cdots\!04\)\( T_{7}^{189} + \)\(48\!\cdots\!52\)\( T_{7}^{188} - \)\(42\!\cdots\!92\)\( T_{7}^{187} - \)\(33\!\cdots\!20\)\( T_{7}^{186} - \)\(79\!\cdots\!12\)\( T_{7}^{185} - \)\(38\!\cdots\!13\)\( T_{7}^{184} + \)\(14\!\cdots\!88\)\( T_{7}^{183} + \)\(19\!\cdots\!16\)\( T_{7}^{182} + \)\(25\!\cdots\!24\)\( T_{7}^{181} + \)\(17\!\cdots\!80\)\( T_{7}^{180} - \)\(40\!\cdots\!32\)\( T_{7}^{179} - \)\(62\!\cdots\!88\)\( T_{7}^{178} - \)\(66\!\cdots\!48\)\( T_{7}^{177} - \)\(61\!\cdots\!58\)\( T_{7}^{176} + \)\(88\!\cdots\!20\)\( T_{7}^{175} + \)\(12\!\cdots\!60\)\( T_{7}^{174} + \)\(13\!\cdots\!00\)\( T_{7}^{173} + \)\(16\!\cdots\!81\)\( T_{7}^{172} - \)\(14\!\cdots\!24\)\( T_{7}^{171} - \)\(16\!\cdots\!56\)\( T_{7}^{170} - \)\(20\!\cdots\!84\)\( T_{7}^{169} - \)\(31\!\cdots\!31\)\( T_{7}^{168} + \)\(18\!\cdots\!36\)\( T_{7}^{167} + \)\(11\!\cdots\!72\)\( T_{7}^{166} + \)\(22\!\cdots\!60\)\( T_{7}^{165} + \)\(45\!\cdots\!23\)\( T_{7}^{164} - \)\(16\!\cdots\!00\)\( T_{7}^{163} + \)\(11\!\cdots\!08\)\( T_{7}^{162} - \)\(16\!\cdots\!44\)\( T_{7}^{161} - \)\(47\!\cdots\!17\)\( T_{7}^{160} + \)\(10\!\cdots\!48\)\( T_{7}^{159} - \)\(10\!\cdots\!20\)\( T_{7}^{158} + \)\(55\!\cdots\!28\)\( T_{7}^{157} + \)\(30\!\cdots\!64\)\( T_{7}^{156} - \)\(42\!\cdots\!64\)\( T_{7}^{155} + \)\(60\!\cdots\!64\)\( T_{7}^{154} - \)\(90\!\cdots\!80\)\( T_{7}^{153} - \)\(91\!\cdots\!74\)\( T_{7}^{152} + \)\(40\!\cdots\!92\)\( T_{7}^{151} + \)\(43\!\cdots\!92\)\( T_{7}^{150} + \)\(48\!\cdots\!84\)\( T_{7}^{149} + \)\(51\!\cdots\!68\)\( T_{7}^{148} - \)\(71\!\cdots\!08\)\( T_{7}^{147} + \)\(35\!\cdots\!28\)\( T_{7}^{146} - \)\(74\!\cdots\!84\)\( T_{7}^{145} - \)\(12\!\cdots\!69\)\( T_{7}^{144} + \)\(70\!\cdots\!88\)\( T_{7}^{143} - \)\(84\!\cdots\!08\)\( T_{7}^{142} + \)\(46\!\cdots\!08\)\( T_{7}^{141} + \)\(15\!\cdots\!10\)\( T_{7}^{140} - \)\(43\!\cdots\!84\)\( T_{7}^{139} + \)\(69\!\cdots\!32\)\( T_{7}^{138} - \)\(12\!\cdots\!92\)\( T_{7}^{137} - \)\(10\!\cdots\!29\)\( T_{7}^{136} + \)\(21\!\cdots\!16\)\( T_{7}^{135} - \)\(27\!\cdots\!00\)\( T_{7}^{134} + \)\(18\!\cdots\!44\)\( T_{7}^{133} + \)\(47\!\cdots\!13\)\( T_{7}^{132} - \)\(11\!\cdots\!80\)\( T_{7}^{131} + \)\(98\!\cdots\!76\)\( T_{7}^{130} - \)\(16\!\cdots\!52\)\( T_{7}^{129} - \)\(14\!\cdots\!45\)\( T_{7}^{128} + \)\(51\!\cdots\!12\)\( T_{7}^{127} - \)\(54\!\cdots\!52\)\( T_{7}^{126} + \)\(90\!\cdots\!96\)\( T_{7}^{125} + \)\(27\!\cdots\!92\)\( T_{7}^{124} - \)\(11\!\cdots\!88\)\( T_{7}^{123} + \)\(22\!\cdots\!96\)\( T_{7}^{122} - \)\(13\!\cdots\!60\)\( T_{7}^{121} - \)\(70\!\cdots\!18\)\( T_{7}^{120} + \)\(12\!\cdots\!08\)\( T_{7}^{119} - \)\(22\!\cdots\!80\)\( T_{7}^{118} + \)\(59\!\cdots\!68\)\( T_{7}^{117} + \)\(23\!\cdots\!94\)\( T_{7}^{116} - \)\(56\!\cdots\!84\)\( T_{7}^{115} - \)\(41\!\cdots\!04\)\( T_{7}^{114} - \)\(22\!\cdots\!12\)\( T_{7}^{113} - \)\(31\!\cdots\!50\)\( T_{7}^{112} + \)\(11\!\cdots\!40\)\( T_{7}^{111} - \)\(13\!\cdots\!00\)\( T_{7}^{110} + \)\(54\!\cdots\!32\)\( T_{7}^{109} - \)\(28\!\cdots\!06\)\( T_{7}^{108} + \)\(17\!\cdots\!68\)\( T_{7}^{107} - \)\(13\!\cdots\!84\)\( T_{7}^{106} - \)\(12\!\cdots\!80\)\( T_{7}^{105} - \)\(14\!\cdots\!51\)\( T_{7}^{104} - \)\(20\!\cdots\!52\)\( T_{7}^{103} + \)\(14\!\cdots\!16\)\( T_{7}^{102} + \)\(15\!\cdots\!68\)\( T_{7}^{101} + \)\(26\!\cdots\!67\)\( T_{7}^{100} - \)\(88\!\cdots\!16\)\( T_{7}^{99} - \)\(17\!\cdots\!64\)\( T_{7}^{98} - \)\(15\!\cdots\!76\)\( T_{7}^{97} + \)\(19\!\cdots\!81\)\( T_{7}^{96} + \)\(12\!\cdots\!40\)\( T_{7}^{95} + \)\(61\!\cdots\!32\)\( T_{7}^{94} + \)\(75\!\cdots\!36\)\( T_{7}^{93} - \)\(57\!\cdots\!02\)\( T_{7}^{92} - \)\(37\!\cdots\!88\)\( T_{7}^{91} - \)\(42\!\cdots\!72\)\( T_{7}^{90} + \)\(97\!\cdots\!40\)\( T_{7}^{89} + \)\(24\!\cdots\!92\)\( T_{7}^{88} + \)\(42\!\cdots\!24\)\( T_{7}^{87} + \)\(23\!\cdots\!72\)\( T_{7}^{86} - \)\(10\!\cdots\!84\)\( T_{7}^{85} + \)\(26\!\cdots\!69\)\( T_{7}^{84} - \)\(16\!\cdots\!20\)\( T_{7}^{83} + \)\(25\!\cdots\!84\)\( T_{7}^{82} + \)\(16\!\cdots\!84\)\( T_{7}^{81} + \)\(23\!\cdots\!07\)\( T_{7}^{80} + \)\(85\!\cdots\!76\)\( T_{7}^{79} - \)\(86\!\cdots\!40\)\( T_{7}^{78} + \)\(88\!\cdots\!72\)\( T_{7}^{77} - \)\(18\!\cdots\!98\)\( T_{7}^{76} + \)\(14\!\cdots\!72\)\( T_{7}^{75} + \)\(11\!\cdots\!60\)\( T_{7}^{74} - \)\(26\!\cdots\!40\)\( T_{7}^{73} + \)\(70\!\cdots\!35\)\( T_{7}^{72} - \)\(12\!\cdots\!56\)\( T_{7}^{71} + \)\(13\!\cdots\!12\)\( T_{7}^{70} - \)\(10\!\cdots\!96\)\( T_{7}^{69} + \)\(25\!\cdots\!08\)\( T_{7}^{68} + \)\(73\!\cdots\!36\)\( T_{7}^{67} - \)\(11\!\cdots\!00\)\( T_{7}^{66} + \)\(10\!\cdots\!96\)\( T_{7}^{65} - \)\(57\!\cdots\!42\)\( T_{7}^{64} - \)\(16\!\cdots\!08\)\( T_{7}^{63} + \)\(68\!\cdots\!60\)\( T_{7}^{62} - \)\(97\!\cdots\!16\)\( T_{7}^{61} + \)\(10\!\cdots\!81\)\( T_{7}^{60} - \)\(11\!\cdots\!60\)\( T_{7}^{59} + \)\(12\!\cdots\!00\)\( T_{7}^{58} - \)\(11\!\cdots\!72\)\( T_{7}^{57} + \)\(99\!\cdots\!20\)\( T_{7}^{56} - \)\(79\!\cdots\!80\)\( T_{7}^{55} + \)\(64\!\cdots\!36\)\( T_{7}^{54} - \)\(50\!\cdots\!76\)\( T_{7}^{53} + \)\(37\!\cdots\!77\)\( T_{7}^{52} - \)\(26\!\cdots\!08\)\( T_{7}^{51} + \)\(17\!\cdots\!24\)\( T_{7}^{50} - \)\(12\!\cdots\!08\)\( T_{7}^{49} + \)\(81\!\cdots\!37\)\( T_{7}^{48} - \)\(51\!\cdots\!32\)\( T_{7}^{47} + \)\(31\!\cdots\!88\)\( T_{7}^{46} - \)\(19\!\cdots\!00\)\( T_{7}^{45} + \)\(11\!\cdots\!14\)\( T_{7}^{44} - \)\(64\!\cdots\!72\)\( T_{7}^{43} + \)\(36\!\cdots\!96\)\( T_{7}^{42} - \)\(19\!\cdots\!76\)\( T_{7}^{41} + \)\(10\!\cdots\!41\)\( T_{7}^{40} - \)\(50\!\cdots\!40\)\( T_{7}^{39} + \)\(24\!\cdots\!64\)\( T_{7}^{38} - \)\(12\!\cdots\!88\)\( T_{7}^{37} + \)\(55\!\cdots\!64\)\( T_{7}^{36} - \)\(21\!\cdots\!36\)\( T_{7}^{35} + \)\(79\!\cdots\!64\)\( T_{7}^{34} - \)\(29\!\cdots\!44\)\( T_{7}^{33} + \)\(99\!\cdots\!08\)\( T_{7}^{32} - \)\(22\!\cdots\!20\)\( T_{7}^{31} + \)\(29\!\cdots\!80\)\( T_{7}^{30} - \)\(10\!\cdots\!12\)\( T_{7}^{29} + \)\(75\!\cdots\!52\)\( T_{7}^{28} - \)\(11\!\cdots\!84\)\( T_{7}^{27} - \)\(13\!\cdots\!60\)\( T_{7}^{26} + \)\(12\!\cdots\!28\)\( T_{7}^{25} - \)\(63\!\cdots\!20\)\( T_{7}^{24} + \)\(26\!\cdots\!84\)\( T_{7}^{23} - \)\(85\!\cdots\!68\)\( T_{7}^{22} + \)\(23\!\cdots\!84\)\( T_{7}^{21} - \)\(56\!\cdots\!44\)\( T_{7}^{20} + \)\(91\!\cdots\!64\)\( T_{7}^{19} + \)\(21\!\cdots\!08\)\( T_{7}^{18} - \)\(22\!\cdots\!92\)\( T_{7}^{17} + \)\(88\!\cdots\!76\)\( T_{7}^{16} - \)\(31\!\cdots\!36\)\( T_{7}^{15} + \)\(12\!\cdots\!36\)\( T_{7}^{14} - \)\(41\!\cdots\!24\)\( T_{7}^{13} + \)\(12\!\cdots\!60\)\( T_{7}^{12} - \)\(33\!\cdots\!68\)\( T_{7}^{11} + \)\(81\!\cdots\!72\)\( T_{7}^{10} - \)\(20\!\cdots\!68\)\( T_{7}^{9} + \)\(47\!\cdots\!28\)\( T_{7}^{8} - \)\(87\!\cdots\!44\)\( T_{7}^{7} + \)\(11\!\cdots\!32\)\( T_{7}^{6} - \)\(34\!\cdots\!64\)\( T_{7}^{5} - \)\(15\!\cdots\!32\)\( T_{7}^{4} + \)\(63\!\cdots\!40\)\( T_{7}^{3} + \)\(20\!\cdots\!92\)\( T_{7}^{2} + \)\(28\!\cdots\!88\)\( T_{7} + \)\(19\!\cdots\!16\)\( \)">\(T_{7}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).