Embedded newform 735.2.y.i.557.12

• Label: 735.2.y.i.557.12
• Level: $735$
• Weight: $2$
• Character: 735.557
• 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			557.12
Character	\(\chi\)	\(=\)	735.557
Dual form			735.2.y.i.128.12

$q$-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 \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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	0.964089	−	0.265578i	\(-0.0855629\pi\)
							0.702137	+	0.712042i	\(0.252230\pi\)
\(3\)	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	−0.0423788	−	0.999102i	\(-0.513494\pi\)
−0.886437	+	0.462850i	\(0.846827\pi\)
\(13\)	−1.28412	+	1.28412i	−0.356151	+	0.356151i	−0.862392	−	0.506241i	\(-0.831035\pi\)
							0.506241	+	0.862392i	\(0.331035\pi\)
\(17\)	−0.792145	−	2.95633i	−0.192123	−	0.717015i	−0.992993	−	0.118175i	\(-0.962296\pi\)
							0.800869	−	0.598839i	\(-0.204371\pi\)
\(19\)	−0.331717	+	0.191517i	−0.0761011	+	0.0439370i	−0.537568	−	0.843221i	\(-0.680657\pi\)
							0.461467	+	0.887158i	\(0.347323\pi\)
\(23\)	−0.658656	+	2.45814i	−0.137339	+	0.512557i	0.862638	+	0.505822i	\(0.168811\pi\)
							−0.999977	+	0.00673550i	\(0.997856\pi\)
\(29\)	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	−0.893652	−	0.448760i	\(-0.851866\pi\)
							0.835464	+	0.549546i	\(0.185199\pi\)
\(37\)	−1.34101	+	5.00473i	−0.220461	+	0.822772i	0.763711	+	0.645558i	\(0.223375\pi\)
							−0.984172	+	0.177214i	\(0.943291\pi\)
\(41\)		−	10.1075i		−	1.57852i	−0.614060	−	0.789259i	\(-0.710465\pi\)
							0.614060	−	0.789259i	\(-0.289535\pi\)
\(43\)	−0.335236	+	0.335236i	−0.0511231	+	0.0511231i	−0.732206	−	0.681083i	\(-0.761509\pi\)
							0.681083	+	0.732206i	\(0.261509\pi\)
\(47\)	−2.80533	−	0.751687i	−0.409200	−	0.109645i	0.0483463	−	0.998831i	\(-0.484605\pi\)
							−0.457546	+	0.889186i	\(0.651272\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.1	✓	48	7.5	odd	6
105.2.x.a.2.12	yes	48	21.5	even	6
105.2.x.a.23.1	yes	48	35.33	even	12
105.2.x.a.23.12	yes	48	105.68	odd	12
105.2.x.a.32.1	yes	48	21.20	even	2
105.2.x.a.32.12	yes	48	7.6	odd	2
105.2.x.a.53.1	yes	48	105.83	odd	4
105.2.x.a.53.12	yes	48	35.13	even	4
525.2.bf.f.32.1		48	35.34	odd	2
525.2.bf.f.32.12		48	105.104	even	2
525.2.bf.f.107.1		48	105.89	even	6
525.2.bf.f.107.12		48	35.19	odd	6
525.2.bf.f.368.1		48	35.27	even	4
525.2.bf.f.368.12		48	105.62	odd	4
525.2.bf.f.443.1		48	105.47	odd	12
525.2.bf.f.443.12		48	35.12	even	12
735.2.j.e.197.1		24	7.4	even	3
735.2.j.e.197.12		24	21.11	odd	6
735.2.j.e.638.1		24	105.53	even	12
735.2.j.e.638.12		24	35.18	odd	12
735.2.j.g.197.1		24	7.3	odd	6
735.2.j.g.197.12		24	21.17	even	6
735.2.j.g.638.1		24	105.38	odd	12
735.2.j.g.638.12		24	35.3	even	12
735.2.y.i.128.1		48	35.23	odd	12	inner
735.2.y.i.128.12		48	105.23	even	12	inner
735.2.y.i.263.1		48	15.8	even	4	inner
735.2.y.i.263.12		48	5.3	odd	4	inner
735.2.y.i.422.1		48	7.2	even	3	inner
735.2.y.i.422.12		48	21.2	odd	6	inner
735.2.y.i.557.1		48	3.2	odd	2	inner
735.2.y.i.557.12		48	1.1	even	1	trivial