Newform orbit 690.2.n.b

• Label: 690.2.n.b
• Level: $690$
• Weight: $2$
• Character orbit: 690.n
• 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.n (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.50967773947\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{2} - 2q^{3} - 24q^{4} + 2q^{6} + 24q^{8} - 6q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
89.1	0.142315	+	0.989821i	−1.73163	−	0.0381696i	−0.959493	+	0.281733i	−2.23565	+	0.0433117i	−0.208656	−	1.71944i	1.95813	−	1.25842i	−0.415415	−	0.909632i	2.99709	+	0.132191i	−0.361037	−	2.20673i
89.2	0.142315	+	0.989821i	−1.67640	−	0.435517i	−0.959493	+	0.281733i	0.584056	+	2.15844i	0.192507	−	1.72132i	−1.28454	+	0.825522i	−0.415415	−	0.909632i	2.62065	+	1.46020i	−2.05335	+	0.885290i
89.3	0.142315	+	0.989821i	−1.67505	+	0.440696i	−0.959493	+	0.281733i	1.25961	+	1.84754i	−0.674594	−	1.59528i	3.27194	−	2.10275i	−0.415415	−	0.909632i	2.61157	−	1.47637i	−1.64947	+	1.50972i
89.4	0.142315	+	0.989821i	−1.56929	−	0.733032i	−0.959493	+	0.281733i	1.76486	−	1.37305i	0.502238	−	1.65764i	−3.04110	+	1.95439i	−0.415415	−	0.909632i	1.92533	+	2.30068i	1.61024	+	1.55149i
89.5	0.142315	+	0.989821i	−1.53419	+	0.803900i	−0.959493	+	0.281733i	−2.02447	−	0.949486i	−1.01406	−	1.40417i	−4.08214	+	2.62343i	−0.415415	−	0.909632i	1.70749	−	2.46667i	0.651709	−	2.13899i
89.6	0.142315	+	0.989821i	−1.27489	+	1.17245i	−0.959493	+	0.281733i	−0.182307	−	2.22862i	−1.34195	−	1.09506i	1.96759	−	1.26449i	−0.415415	−	0.909632i	0.250714	−	2.98951i	2.17999	−	0.497618i
89.7	0.142315	+	0.989821i	−1.11013	+	1.32951i	−0.959493	+	0.281733i	1.79300	−	1.33609i	−1.47397	−	0.909626i	−0.483302	+	0.310599i	−0.415415	−	0.909632i	−0.535201	−	2.95187i	1.57766	+	1.58461i
89.8	0.142315	+	0.989821i	−0.804725	−	1.53376i	−0.959493	+	0.281733i	−2.06421	−	0.859668i	1.40362	−	1.01481i	0.955977	−	0.614369i	−0.415415	−	0.909632i	−1.70483	+	2.46851i	0.557150	−	2.16554i
89.9	0.142315	+	0.989821i	−0.748393	−	1.56202i	−0.959493	+	0.281733i	−1.26417	+	1.84441i	1.43961	−	0.963074i	−1.66558	+	1.07040i	−0.415415	−	0.909632i	−1.87982	+	2.33801i	−2.00555	−	0.988819i
89.10	0.142315	+	0.989821i	−0.730817	−	1.57032i	−0.959493	+	0.281733i	2.23496	−	0.0702463i	1.45033	−	0.946859i	3.21136	−	2.06381i	−0.415415	−	0.909632i	−1.93181	+	2.29523i	0.387600	+	2.20222i
89.11	0.142315	+	0.989821i	−0.277794	+	1.70963i	−0.959493	+	0.281733i	−1.06732	+	1.96490i	−1.73176	−	0.0316608i	0.483302	−	0.310599i	−0.415415	−	0.909632i	−2.84566	−	0.949849i	−2.09680	−	0.776821i
89.12	0.142315	+	0.989821i	−0.0575173	−	1.73110i	−0.959493	+	0.281733i	2.02703	+	0.944001i	1.70529	−	0.303292i	−1.08447	+	0.696948i	−0.415415	−	0.909632i	−2.99338	+	0.199136i	−0.645916	+	2.14075i
89.13	0.142315	+	0.989821i	−0.0512012	+	1.73129i	−0.959493	+	0.281733i	−2.23188	+	0.136714i	−1.72096	+	0.195709i	−1.96759	+	1.26449i	−0.415415	−	0.909632i	−2.99476	−	0.177289i	−0.452953	−	2.18971i
89.14	0.142315	+	0.989821i	0.397135	+	1.68591i	−0.959493	+	0.281733i	−1.22793	−	1.86874i	−1.61223	+	0.633022i	4.08214	−	2.62343i	−0.415415	−	0.909632i	−2.68457	+	1.33906i	1.67496	−	1.48138i
89.15	0.142315	+	0.989821i	0.479764	−	1.66428i	−0.959493	+	0.281733i	0.0269160	−	2.23591i	1.71562	+	0.238029i	−2.74772	+	1.76585i	−0.415415	−	0.909632i	−2.53965	−	1.59692i	2.21698	−	0.291561i
89.16	0.142315	+	0.989821i	0.763868	+	1.55451i	−0.959493	+	0.281733i	2.00799	+	0.983855i	−1.42998	+	0.977323i	−3.27194	+	2.10275i	−0.415415	−	0.909632i	−1.83301	+	2.37488i	−0.688074	+	2.12757i
89.17	0.142315	+	0.989821i	0.943600	−	1.45245i	−0.959493	+	0.281733i	−2.20932	+	0.344845i	1.57196	+	0.727290i	2.74772	−	1.76585i	−0.415415	−	0.909632i	−1.21924	−	2.74107i	−0.655753	−	2.13775i
89.18	0.142315	+	0.989821i	1.16282	+	1.28368i	−0.959493	+	0.281733i	−0.275295	−	2.21906i	−1.10513	+	1.33367i	−1.95813	+	1.25842i	−0.415415	−	0.909632i	−0.295684	+	2.98539i	2.15729	−	0.588298i
89.19	0.142315	+	0.989821i	1.34594	−	1.09016i	−0.959493	+	0.281733i	1.22287	+	1.87206i	1.27061	+	1.17710i	1.08447	−	0.696948i	−0.415415	−	0.909632i	0.623112	−	2.93458i	−1.67897	+	1.47684i
89.20	0.142315	+	0.989821i	1.42695	+	0.981738i	−0.959493	+	0.281733i	2.21959	+	0.270933i	−0.768669	+	1.55214i	1.28454	−	0.825522i	−0.415415	−	0.909632i	1.07238	+	2.80178i	0.0477058	+	2.23556i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	inner
23.d	odd	22	1	inner
345.n	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.n.b	yes	240
3.b	odd	2	1		690.2.n.a	✓	240
5.b	even	2	1		690.2.n.a	✓	240
15.d	odd	2	1	inner	690.2.n.b	yes	240
23.d	odd	22	1	inner	690.2.n.b	yes	240
69.g	even	22	1		690.2.n.a	✓	240
115.i	odd	22	1		690.2.n.a	✓	240
345.n	even	22	1	inner	690.2.n.b	yes	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.n.a	✓	240	3.b	odd	2	1
690.2.n.a	✓	240	5.b	even	2	1
690.2.n.a	✓	240	69.g	even	22	1
690.2.n.a	✓	240	115.i	odd	22	1
690.2.n.b	yes	240	1.a	even	1	1	trivial
690.2.n.b	yes	240	15.d	odd	2	1	inner
690.2.n.b	yes	240	23.d	odd	22	1	inner
690.2.n.b	yes	240	345.n	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(19\!\cdots\!51\)\( T_{17}^{106} - \)\(15\!\cdots\!27\)\( T_{17}^{105} + \)\(96\!\cdots\!76\)\( T_{17}^{104} + \)\(61\!\cdots\!89\)\( T_{17}^{103} - \)\(51\!\cdots\!64\)\( T_{17}^{102} - \)\(18\!\cdots\!44\)\( T_{17}^{101} + \)\(29\!\cdots\!06\)\( T_{17}^{100} + \)\(94\!\cdots\!68\)\( T_{17}^{99} - \)\(99\!\cdots\!68\)\( T_{17}^{98} - \)\(27\!\cdots\!78\)\( T_{17}^{97} + \)\(33\!\cdots\!44\)\( T_{17}^{96} + \)\(59\!\cdots\!28\)\( T_{17}^{95} - \)\(12\!\cdots\!73\)\( T_{17}^{94} - \)\(15\!\cdots\!25\)\( T_{17}^{93} + \)\(45\!\cdots\!91\)\( T_{17}^{92} + \)\(11\!\cdots\!98\)\( T_{17}^{91} - \)\(97\!\cdots\!90\)\( T_{17}^{90} - \)\(22\!\cdots\!98\)\( T_{17}^{89} + \)\(23\!\cdots\!08\)\( T_{17}^{88} - \)\(16\!\cdots\!86\)\( T_{17}^{87} - \)\(12\!\cdots\!08\)\( T_{17}^{86} - \)\(14\!\cdots\!88\)\( T_{17}^{85} + \)\(39\!\cdots\!78\)\( T_{17}^{84} + \)\(12\!\cdots\!03\)\( T_{17}^{83} - \)\(74\!\cdots\!42\)\( T_{17}^{82} - \)\(40\!\cdots\!52\)\( T_{17}^{81} + \)\(98\!\cdots\!41\)\( T_{17}^{80} + \)\(11\!\cdots\!36\)\( T_{17}^{79} + \)\(24\!\cdots\!16\)\( T_{17}^{78} + \)\(23\!\cdots\!29\)\( T_{17}^{77} + \)\(77\!\cdots\!13\)\( T_{17}^{76} - \)\(16\!\cdots\!07\)\( T_{17}^{75} - \)\(14\!\cdots\!89\)\( T_{17}^{74} + \)\(17\!\cdots\!30\)\( T_{17}^{73} + \)\(60\!\cdots\!76\)\( T_{17}^{72} + \)\(28\!\cdots\!25\)\( T_{17}^{71} + \)\(39\!\cdots\!24\)\( T_{17}^{70} - \)\(11\!\cdots\!49\)\( T_{17}^{69} - \)\(72\!\cdots\!25\)\( T_{17}^{68} - \)\(28\!\cdots\!13\)\( T_{17}^{67} - \)\(12\!\cdots\!96\)\( T_{17}^{66} - \)\(34\!\cdots\!22\)\( T_{17}^{65} - \)\(47\!\cdots\!65\)\( T_{17}^{64} - \)\(35\!\cdots\!04\)\( T_{17}^{63} - \)\(31\!\cdots\!09\)\( T_{17}^{62} - \)\(72\!\cdots\!50\)\( T_{17}^{61} + \)\(37\!\cdots\!82\)\( T_{17}^{60} + \)\(31\!\cdots\!72\)\( T_{17}^{59} + \)\(10\!\cdots\!91\)\( T_{17}^{58} + \)\(24\!\cdots\!42\)\( T_{17}^{57} + \)\(43\!\cdots\!32\)\( T_{17}^{56} + \)\(24\!\cdots\!33\)\( T_{17}^{55} - \)\(55\!\cdots\!95\)\( T_{17}^{54} - \)\(39\!\cdots\!78\)\( T_{17}^{53} - \)\(15\!\cdots\!93\)\( T_{17}^{52} - \)\(36\!\cdots\!39\)\( T_{17}^{51} - \)\(56\!\cdots\!82\)\( T_{17}^{50} - \)\(10\!\cdots\!15\)\( T_{17}^{49} + \)\(38\!\cdots\!96\)\( T_{17}^{48} + \)\(22\!\cdots\!78\)\( T_{17}^{47} + \)\(86\!\cdots\!97\)\( T_{17}^{46} + \)\(26\!\cdots\!57\)\( T_{17}^{45} + \)\(68\!\cdots\!57\)\( T_{17}^{44} + \)\(15\!\cdots\!98\)\( T_{17}^{43} + \)\(35\!\cdots\!29\)\( T_{17}^{42} + \)\(76\!\cdots\!92\)\( T_{17}^{41} + \)\(15\!\cdots\!72\)\( T_{17}^{40} + \)\(31\!\cdots\!57\)\( T_{17}^{39} + \)\(60\!\cdots\!65\)\( T_{17}^{38} + \)\(10\!\cdots\!58\)\( T_{17}^{37} + \)\(17\!\cdots\!35\)\( T_{17}^{36} + \)\(28\!\cdots\!32\)\( T_{17}^{35} + \)\(52\!\cdots\!97\)\( T_{17}^{34} + \)\(10\!\cdots\!21\)\( T_{17}^{33} + \)\(20\!\cdots\!63\)\( T_{17}^{32} + \)\(33\!\cdots\!42\)\( T_{17}^{31} + \)\(44\!\cdots\!93\)\( T_{17}^{30} + \)\(51\!\cdots\!10\)\( T_{17}^{29} + \)\(55\!\cdots\!10\)\( T_{17}^{28} + \)\(59\!\cdots\!76\)\( T_{17}^{27} + \)\(58\!\cdots\!68\)\( T_{17}^{26} + \)\(47\!\cdots\!60\)\( T_{17}^{25} + \)\(36\!\cdots\!52\)\( T_{17}^{24} + \)\(39\!\cdots\!08\)\( T_{17}^{23} + \)\(51\!\cdots\!60\)\( T_{17}^{22} + \)\(54\!\cdots\!52\)\( T_{17}^{21} + \)\(43\!\cdots\!64\)\( T_{17}^{20} + \)\(34\!\cdots\!20\)\( T_{17}^{19} + \)\(34\!\cdots\!72\)\( T_{17}^{18} + \)\(27\!\cdots\!84\)\( T_{17}^{17} + \)\(11\!\cdots\!92\)\( T_{17}^{16} - \)\(17\!\cdots\!60\)\( T_{17}^{15} - \)\(26\!\cdots\!76\)\( T_{17}^{14} + \)\(81\!\cdots\!08\)\( T_{17}^{13} + \)\(12\!\cdots\!00\)\( T_{17}^{12} + \)\(52\!\cdots\!96\)\( T_{17}^{11} - \)\(19\!\cdots\!84\)\( T_{17}^{10} - \)\(69\!\cdots\!88\)\( T_{17}^{9} + \)\(20\!\cdots\!92\)\( T_{17}^{8} + \)\(11\!\cdots\!16\)\( T_{17}^{7} + \)\(76\!\cdots\!36\)\( T_{17}^{6} - \)\(36\!\cdots\!88\)\( T_{17}^{5} - \)\(12\!\cdots\!12\)\( T_{17}^{4} + \)\(16\!\cdots\!44\)\( T_{17}^{3} + \)\(57\!\cdots\!72\)\( T_{17}^{2} + \)\(39\!\cdots\!56\)\( T_{17} + \)\(23\!\cdots\!84\)\( \)">\(T_{17}^{120} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).