Newform orbit 90.6.i.a

• Label: 90.6.i.a
• Level: $90$
• Weight: $6$
• Character orbit: 90.i
• Analytic conductor: $14.435$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	90.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.4345437832\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 480 q^{4} + 58 q^{5} - 80 q^{6} - 754 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1	−3.46410	+	2.00000i	−15.4255	+	2.24788i	8.00000	−	13.8564i	1.83828	−	55.8715i	48.9398	−	38.6380i	11.0110	−	6.35720i			64.0000i	232.894	−	69.3496i	105.375	+	197.221i
49.2	−3.46410	+	2.00000i	−13.3044	−	8.12364i	8.00000	−	13.8564i	−24.4045	+	50.2933i	62.3350	+	1.53238i	44.2668	−	25.5575i			64.0000i	111.013	+	216.160i	−16.0469	−	223.030i
49.3	−3.46410	+	2.00000i	−12.7147	−	9.01871i	8.00000	−	13.8564i	55.8971	+	0.720288i	62.0823	+	5.81241i	−142.301	+	82.1573i			64.0000i	80.3256	+	229.340i	−195.074	+	109.299i
49.4	−3.46410	+	2.00000i	−11.6601	+	10.3462i	8.00000	−	13.8564i	0.721783	+	55.8970i	19.6993	−	59.1603i	165.478	−	95.5387i			64.0000i	28.9139	−	241.274i	−114.294	−	192.189i
49.5	−3.46410	+	2.00000i	−9.77477	+	12.1431i	8.00000	−	13.8564i	−55.8921	+	1.03828i	9.57468	−	61.6143i	−108.162	+	62.4473i			64.0000i	−51.9077	−	237.391i	191.539	−	115.381i
49.6	−3.46410	+	2.00000i	−3.03447	+	15.2903i	8.00000	−	13.8564i	51.1909	+	22.4609i	−20.0688	−	59.0360i	−126.581	+	73.0819i			64.0000i	−224.584	−	92.7957i	−222.252	+	24.5749i
49.7	−3.46410	+	2.00000i	−1.07994	−	15.5510i	8.00000	−	13.8564i	49.7460	−	25.5017i	34.8430	+	51.7104i	197.128	−	113.812i			64.0000i	−240.667	+	33.5882i	−121.322	+	187.832i
49.8	−3.46410	+	2.00000i	−0.656298	−	15.5746i	8.00000	−	13.8564i	−50.6476	−	23.6606i	33.4228	+	52.6395i	−133.336	+	76.9815i			64.0000i	−242.139	+	20.4432i	222.770	−	19.3323i
49.9	−3.46410	+	2.00000i	2.22547	+	15.4288i	8.00000	−	13.8564i	44.1872	−	34.2416i	−38.5668	−	48.9959i	104.030	−	60.0617i			64.0000i	−233.095	+	68.6727i	−84.5858	+	206.991i
49.10	−3.46410	+	2.00000i	5.33576	−	14.6468i	8.00000	−	13.8564i	9.35769	+	55.1129i	10.8100	+	61.4096i	−47.4037	+	27.3686i			64.0000i	−186.059	−	156.304i	−142.642	−	172.201i
49.11	−3.46410	+	2.00000i	7.21689	+	13.8173i	8.00000	−	13.8564i	−55.4377	−	7.18779i	−52.6345	−	33.4306i	206.233	−	119.069i			64.0000i	−138.833	+	199.435i	206.417	−	85.9761i
49.12	−3.46410	+	2.00000i	12.8499	−	8.82495i	8.00000	−	13.8564i	−12.5843	−	54.4668i	−26.8635	+	56.2703i	76.1068	−	43.9403i			64.0000i	87.2406	−	226.800i	152.527	+	163.510i
49.13	−3.46410	+	2.00000i	13.3751	+	8.00671i	8.00000	−	13.8564i	9.26098	−	55.1292i	−62.3460	−	0.985922i	−133.363	+	76.9972i			64.0000i	114.785	+	214.181i	78.1775	+	209.495i
49.14	−3.46410	+	2.00000i	15.4015	−	2.40672i	8.00000	−	13.8564i	−54.3448	+	13.1011i	−48.5391	+	39.1402i	6.35690	−	3.67016i			64.0000i	231.415	−	74.1344i	162.054	−	154.073i
49.15	−3.46410	+	2.00000i	15.5756	−	0.633612i	8.00000	−	13.8564i	45.6111	+	32.3207i	−52.6882	+	33.3460i	45.9476	−	26.5278i			64.0000i	242.197	−	19.7377i	−222.643	−	20.7401i
49.16	3.46410	−	2.00000i	−15.5756	+	0.633612i	8.00000	−	13.8564i	−50.7961	−	23.3400i	−52.6882	+	33.3460i	−45.9476	+	26.5278i		−	64.0000i	242.197	−	19.7377i	−222.643	+	20.7401i
49.17	3.46410	−	2.00000i	−15.4015	+	2.40672i	8.00000	−	13.8564i	15.8266	+	53.6145i	−48.5391	+	39.1402i	−6.35690	+	3.67016i		−	64.0000i	231.415	−	74.1344i	162.054	+	154.073i
49.18	3.46410	−	2.00000i	−13.3751	−	8.00671i	8.00000	−	13.8564i	43.1128	−	35.5849i	−62.3460	−	0.985922i	133.363	−	76.9972i		−	64.0000i	114.785	+	214.181i	78.1775	−	209.495i
49.19	3.46410	−	2.00000i	−12.8499	+	8.82495i	8.00000	−	13.8564i	53.4618	−	16.3351i	−26.8635	+	56.2703i	−76.1068	+	43.9403i		−	64.0000i	87.2406	−	226.800i	152.527	−	163.510i
49.20	3.46410	−	2.00000i	−7.21689	−	13.8173i	8.00000	−	13.8564i	33.9436	+	44.4165i	−52.6345	−	33.4306i	−206.233	+	119.069i		−	64.0000i	−138.833	+	199.435i	206.417	+	85.9761i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	90.6.i.a	✓	60
3.b	odd	2	1		270.6.i.a		60
5.b	even	2	1	inner	90.6.i.a	✓	60
9.c	even	3	1	inner	90.6.i.a	✓	60
9.d	odd	6	1		270.6.i.a		60
15.d	odd	2	1		270.6.i.a		60
45.h	odd	6	1		270.6.i.a		60
45.j	even	6	1	inner	90.6.i.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.6.i.a	✓	60	1.a	even	1	1	trivial
90.6.i.a	✓	60	5.b	even	2	1	inner
90.6.i.a	✓	60	9.c	even	3	1	inner
90.6.i.a	✓	60	45.j	even	6	1	inner
270.6.i.a		60	3.b	odd	2	1
270.6.i.a		60	9.d	odd	6	1
270.6.i.a		60	15.d	odd	2	1
270.6.i.a		60	45.h	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(90, [\chi])\).