Newform orbit 690.2.w.b

• 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: $4$

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	−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

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