Newform orbit 690.2.w.a

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 240q - 24q^{6} + O(q^{10}) \)

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

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