Embedded newform 735.2.y.i.128.12

• Label: 735.2.y.i.128.12
• Level: $735$
• Weight: $2$
• Character: 735.128
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			128.12
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.i.557.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.35640 - 0.631395i) q^{2} +(1.54878 + 0.775426i) q^{3} +(3.42191 - 1.97564i) q^{4} +(0.0540016 - 2.23542i) q^{5} +(4.13914 + 0.849321i) q^{6} +(3.36596 - 3.36596i) q^{8} +(1.79743 + 2.40193i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.35640	−	0.631395i	1.66623	−	0.446464i	0.702137	−	0.712042i	\(-0.252230\pi\)
							0.964089	+	0.265578i	\(0.0855629\pi\)
\(3\)	1.54878	+	0.775426i	0.894188	+	0.447692i
							\(5\)	0.0540016	−	2.23542i	0.0241502	−	0.999708i
\(7\)		0			0
							\(11\)	−3.08053	+	1.77855i	−0.928816	+	0.536252i	−0.886437	−	0.462850i	\(-0.846827\pi\)
−0.0423788	+	0.999102i	\(0.513494\pi\)
\(13\)	−1.28412	−	1.28412i	−0.356151	−	0.356151i	0.506241	−	0.862392i	\(-0.331035\pi\)
							−0.862392	+	0.506241i	\(0.831035\pi\)
\(17\)	−0.792145	+	2.95633i	−0.192123	+	0.717015i	0.800869	+	0.598839i	\(0.204371\pi\)
							−0.992993	+	0.118175i	\(0.962296\pi\)
\(19\)	−0.331717	−	0.191517i	−0.0761011	−	0.0439370i	0.461467	−	0.887158i	\(-0.347323\pi\)
							−0.537568	+	0.843221i	\(0.680657\pi\)
\(23\)	−0.658656	−	2.45814i	−0.137339	−	0.512557i	−0.999977	−	0.00673550i	\(-0.997856\pi\)
							0.862638	−	0.505822i	\(-0.168811\pi\)
\(29\)	5.51741			1.02456			0.512279	−	0.858819i	\(-0.328801\pi\)
							0.512279	+	0.858819i	\(0.328801\pi\)
\(31\)	−0.323980	−	0.561149i	−0.0581885	−	0.100785i	0.835464	−	0.549546i	\(-0.185199\pi\)
							−0.893652	+	0.448760i	\(0.851866\pi\)
\(37\)	−1.34101	−	5.00473i	−0.220461	−	0.822772i	−0.984172	−	0.177214i	\(-0.943291\pi\)
							0.763711	−	0.645558i	\(-0.223375\pi\)
\(41\)			10.1075i			1.57852i	0.614060	+	0.789259i	\(0.289535\pi\)
							−0.614060	+	0.789259i	\(0.710465\pi\)
\(43\)	−0.335236	−	0.335236i	−0.0511231	−	0.0511231i	0.681083	−	0.732206i	\(-0.261509\pi\)
							−0.732206	+	0.681083i	\(0.761509\pi\)
\(47\)	−2.80533	+	0.751687i	−0.409200	+	0.109645i	−0.457546	−	0.889186i	\(-0.651272\pi\)
							0.0483463	+	0.998831i	\(0.484605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.i.128.12		48
3.2	odd	2	inner	735.2.y.i.128.1		48
5.2	odd	4	inner	735.2.y.i.422.12		48
7.2	even	3		735.2.j.e.638.1		24
7.3	odd	6		105.2.x.a.53.1	yes	48
7.4	even	3	inner	735.2.y.i.263.1		48
7.5	odd	6		735.2.j.g.638.1		24
7.6	odd	2		105.2.x.a.23.12	yes	48
15.2	even	4	inner	735.2.y.i.422.1		48
21.2	odd	6		735.2.j.e.638.12		24
21.5	even	6		735.2.j.g.638.12		24
21.11	odd	6	inner	735.2.y.i.263.12		48
21.17	even	6		105.2.x.a.53.12	yes	48
21.20	even	2		105.2.x.a.23.1	yes	48
35.2	odd	12		735.2.j.e.197.12		24
35.3	even	12		525.2.bf.f.32.12		48
35.12	even	12		735.2.j.g.197.12		24
35.13	even	4		525.2.bf.f.107.1		48
35.17	even	12		105.2.x.a.32.1	yes	48
35.24	odd	6		525.2.bf.f.368.12		48
35.27	even	4		105.2.x.a.2.12	yes	48
35.32	odd	12	inner	735.2.y.i.557.1		48
35.34	odd	2		525.2.bf.f.443.1		48
105.2	even	12		735.2.j.e.197.1		24
105.17	odd	12		105.2.x.a.32.12	yes	48
105.32	even	12	inner	735.2.y.i.557.12		48
105.38	odd	12		525.2.bf.f.32.1		48
105.47	odd	12		735.2.j.g.197.1		24
105.59	even	6		525.2.bf.f.368.1		48
105.62	odd	4		105.2.x.a.2.1	✓	48
105.83	odd	4		525.2.bf.f.107.12		48
105.104	even	2		525.2.bf.f.443.12		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.1	✓	48	105.62	odd	4
105.2.x.a.2.12	yes	48	35.27	even	4
105.2.x.a.23.1	yes	48	21.20	even	2
105.2.x.a.23.12	yes	48	7.6	odd	2
105.2.x.a.32.1	yes	48	35.17	even	12
105.2.x.a.32.12	yes	48	105.17	odd	12
105.2.x.a.53.1	yes	48	7.3	odd	6
105.2.x.a.53.12	yes	48	21.17	even	6
525.2.bf.f.32.1		48	105.38	odd	12
525.2.bf.f.32.12		48	35.3	even	12
525.2.bf.f.107.1		48	35.13	even	4
525.2.bf.f.107.12		48	105.83	odd	4
525.2.bf.f.368.1		48	105.59	even	6
525.2.bf.f.368.12		48	35.24	odd	6
525.2.bf.f.443.1		48	35.34	odd	2
525.2.bf.f.443.12		48	105.104	even	2
735.2.j.e.197.1		24	105.2	even	12
735.2.j.e.197.12		24	35.2	odd	12
735.2.j.e.638.1		24	7.2	even	3
735.2.j.e.638.12		24	21.2	odd	6
735.2.j.g.197.1		24	105.47	odd	12
735.2.j.g.197.12		24	35.12	even	12
735.2.j.g.638.1		24	7.5	odd	6
735.2.j.g.638.12		24	21.5	even	6
735.2.y.i.128.1		48	3.2	odd	2	inner
735.2.y.i.128.12		48	1.1	even	1	trivial
735.2.y.i.263.1		48	7.4	even	3	inner
735.2.y.i.263.12		48	21.11	odd	6	inner
735.2.y.i.422.1		48	15.2	even	4	inner
735.2.y.i.422.12		48	5.2	odd	4	inner
735.2.y.i.557.1		48	35.32	odd	12	inner
735.2.y.i.557.12		48	105.32	even	12	inner