Newform orbit 690.2.q.b

• Label: 690.2.q.b
• Level: $690$
• Weight: $2$
• Character orbit: 690.q
• Analytic conductor: $5.510$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

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.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{4} + 16q^{5} - 2q^{6} - 46q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−0.540641	−	0.841254i	−1.68349	−	0.407274i	−0.415415	+	0.909632i	0.959493	−	0.281733i	0.567541	+	1.63643i	0.0904807	−	0.0784019i	0.989821	−	0.142315i	2.66826	+	1.37128i	−0.755750	−	0.654861i
11.2	−0.540641	−	0.841254i	−1.23589	+	1.21350i	−0.415415	+	0.909632i	0.959493	−	0.281733i	1.68903	+	0.383629i	−1.57741	+	1.36683i	0.989821	−	0.142315i	0.0548407	−	2.99950i	−0.755750	−	0.654861i
11.3	−0.540641	−	0.841254i	−1.01936	−	1.40032i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−0.626917	+	1.61461i	2.43170	−	2.10708i	0.989821	−	0.142315i	−0.921804	+	2.85487i	−0.755750	−	0.654861i
11.4	−0.540641	−	0.841254i	−0.329943	+	1.70033i	−0.415415	+	0.909632i	0.959493	−	0.281733i	1.60879	−	0.641705i	0.461235	−	0.399662i	0.989821	−	0.142315i	−2.78228	−	1.12203i	−0.755750	−	0.654861i
11.5	−0.540641	−	0.841254i	−0.0752308	−	1.73042i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.41505	+	0.998822i	−2.94981	+	2.55602i	0.989821	−	0.142315i	−2.98868	+	0.260361i	−0.755750	−	0.654861i
11.6	−0.540641	−	0.841254i	0.869657	+	1.49790i	−0.415415	+	0.909632i	0.959493	−	0.281733i	0.789940	−	1.54143i	3.70222	−	3.20799i	0.989821	−	0.142315i	−1.48739	+	2.60531i	−0.755750	−	0.654861i
11.7	−0.540641	−	0.841254i	0.989774	−	1.42139i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.73086	−	0.0641895i	−0.315364	+	0.273265i	0.989821	−	0.142315i	−1.04070	−	2.81371i	−0.755750	−	0.654861i
11.8	−0.540641	−	0.841254i	1.72873	−	0.107190i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.02480	−	1.39635i	1.12640	−	0.976034i	0.989821	−	0.142315i	2.97702	−	0.370607i	−0.755750	−	0.654861i
11.9	0.540641	+	0.841254i	−1.68374	+	0.406235i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.25204	−	1.19682i	0.920755	−	0.797839i	−0.989821	+	0.142315i	2.66995	−	1.36799i	0.755750	+	0.654861i
11.10	0.540641	+	0.841254i	−1.50413	+	0.858826i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.53569	−	0.801041i	−1.03609	+	0.897775i	−0.989821	+	0.142315i	1.52484	−	2.58358i	0.755750	+	0.654861i
11.11	0.540641	+	0.841254i	−1.16057	−	1.28572i	−0.415415	+	0.909632i	0.959493	−	0.281733i	0.454160	−	1.67145i	−1.93677	+	1.67822i	−0.989821	+	0.142315i	−0.306134	+	2.98434i	0.755750	+	0.654861i
11.12	0.540641	+	0.841254i	0.244221	+	1.71475i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−1.31050	+	1.13251i	3.34933	−	2.90221i	−0.989821	+	0.142315i	−2.88071	+	0.837554i	0.755750	+	0.654861i
11.13	0.540641	+	0.841254i	0.416404	−	1.68125i	−0.415415	+	0.909632i	0.959493	−	0.281733i	1.63948	−	0.558652i	1.57600	−	1.36561i	−0.989821	+	0.142315i	−2.65322	−	1.40016i	0.755750	+	0.654861i
11.14	0.540641	+	0.841254i	1.36664	+	1.06409i	−0.415415	+	0.909632i	0.959493	−	0.281733i	−0.156311	+	1.72498i	−3.41511	+	2.95921i	−0.989821	+	0.142315i	0.735409	+	2.90847i	0.755750	+	0.654861i
11.15	0.540641	+	0.841254i	1.43420	−	0.971113i	−0.415415	+	0.909632i	0.959493	−	0.281733i	1.59234	+	0.681505i	−3.34609	+	2.89940i	−0.989821	+	0.142315i	1.11388	−	2.78555i	0.755750	+	0.654861i
11.16	0.540641	+	0.841254i	1.64273	+	0.549041i	−0.415415	+	0.909632i	0.959493	−	0.281733i	0.426243	+	1.67878i	0.918508	−	0.795892i	−0.989821	+	0.142315i	2.39711	+	1.80385i	0.755750	+	0.654861i
191.1	−0.755750	−	0.654861i	−1.58079	−	0.707884i	0.142315	+	0.989821i	−0.415415	−	0.909632i	0.731117	+	1.57018i	0.325415	+	1.10826i	0.540641	−	0.841254i	1.99780	+	2.23803i	−0.281733	+	0.959493i
191.2	−0.755750	−	0.654861i	−1.30022	+	1.14430i	0.142315	+	0.989821i	−0.415415	−	0.909632i	1.73200	−	0.0133460i	0.519422	+	1.76899i	0.540641	−	0.841254i	0.381134	−	2.97569i	−0.281733	+	0.959493i
191.3	−0.755750	−	0.654861i	−0.982602	+	1.42636i	0.142315	+	0.989821i	−0.415415	−	0.909632i	1.67667	−	0.434500i	−0.819457	−	2.79081i	0.540641	−	0.841254i	−1.06898	−	2.80308i	−0.281733	+	0.959493i
191.4	−0.755750	−	0.654861i	−0.340196	−	1.69831i	0.142315	+	0.989821i	−0.415415	−	0.909632i	−0.855055	+	1.50628i	0.680617	+	2.31797i	0.540641	−	0.841254i	−2.76853	+	1.15552i	−0.281733	+	0.959493i

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.b	yes	160
3.b	odd	2	1		690.2.q.a	✓	160
23.d	odd	22	1		690.2.q.a	✓	160
69.g	even	22	1	inner	690.2.q.b	yes	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.q.a	✓	160	3.b	odd	2	1
690.2.q.a	✓	160	23.d	odd	22	1
690.2.q.b	yes	160	1.a	even	1	1	trivial
690.2.q.b	yes	160	69.g	even	22	1	inner

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