Embedded newform 735.2.y.i.263.2

• Label: 735.2.y.i.263.2
• Level: $735$
• Weight: $2$
• Character: 735.263
• 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			263.2
Character	\(\chi\)	\(=\)	735.263
Dual form			735.2.y.i.422.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.582118 + 2.17249i) q^{2} +(-1.60750 + 0.644934i) q^{3} +(-2.64881 - 1.52929i) q^{4} +(-2.21039 + 0.337883i) q^{5} +(-0.465359 - 3.86771i) q^{6} +(1.68355 - 1.68355i) q^{8} +(2.16812 - 2.07346i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.582118	+	2.17249i	−0.411619	+	1.53618i	0.379893	+	0.925031i	\(0.375961\pi\)
							−0.791512	+	0.611154i	\(0.790706\pi\)
\(3\)	−1.60750	+	0.644934i	−0.928091	+	0.372353i
							\(5\)	−2.21039	+	0.337883i	−0.988518	+	0.151106i
\(7\)		0			0
							\(11\)	−3.88729	−	2.24433i	−1.17206	−	0.676690i	−0.217896	−	0.975972i	\(-0.569919\pi\)
−0.954165	+	0.299282i	\(0.903253\pi\)
\(13\)	1.08424	+	1.08424i	0.300715	+	0.300715i	0.841294	−	0.540578i	\(-0.181795\pi\)
							−0.540578	+	0.841294i	\(0.681795\pi\)
\(17\)	−2.04863	+	0.548929i	−0.496866	+	0.133135i	−0.498545	−	0.866864i	\(-0.666132\pi\)
							0.00167924	+	0.999999i	\(0.499465\pi\)
\(19\)	3.66075	−	2.11354i	0.839834	−	0.484878i	−0.0173737	−	0.999849i	\(-0.505531\pi\)
							0.857208	+	0.514971i	\(0.172197\pi\)
\(23\)	3.13628	+	0.840363i	0.653959	+	0.175228i	0.570518	−	0.821285i	\(-0.306742\pi\)
							0.0834407	+	0.996513i	\(0.473409\pi\)
\(29\)	−1.69118			−0.314045			−0.157023	−	0.987595i	\(-0.550190\pi\)
							−0.157023	+	0.987595i	\(0.550190\pi\)
\(31\)	−0.530077	+	0.918121i	−0.0952047	+	0.164899i	−0.909694	−	0.415279i	\(-0.863684\pi\)
							0.814489	+	0.580179i	\(0.197017\pi\)
\(37\)	−5.75771	−	1.54277i	−0.946562	−	0.253631i	−0.247659	−	0.968847i	\(-0.579661\pi\)
							−0.698903	+	0.715217i	\(0.746328\pi\)
\(41\)			5.84230i			0.912414i	0.889874	+	0.456207i	\(0.150792\pi\)
							−0.889874	+	0.456207i	\(0.849208\pi\)
\(43\)	2.00369	+	2.00369i	0.305559	+	0.305559i	0.843184	−	0.537625i	\(-0.180678\pi\)
							−0.537625	+	0.843184i	\(0.680678\pi\)
\(47\)	1.36662	−	5.10030i	0.199342	−	0.743956i	−0.791758	−	0.610835i	\(-0.790834\pi\)
							0.991100	−	0.133120i	\(-0.0424996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.i.263.2		48
3.2	odd	2	inner	735.2.y.i.263.11		48
5.2	odd	4	inner	735.2.y.i.557.2		48
7.2	even	3	inner	735.2.y.i.128.11		48
7.3	odd	6		735.2.j.g.638.2		24
7.4	even	3		735.2.j.e.638.2		24
7.5	odd	6		105.2.x.a.23.11	yes	48
7.6	odd	2		105.2.x.a.53.2	yes	48
15.2	even	4	inner	735.2.y.i.557.11		48
21.2	odd	6	inner	735.2.y.i.128.2		48
21.5	even	6		105.2.x.a.23.2	yes	48
21.11	odd	6		735.2.j.e.638.11		24
21.17	even	6		735.2.j.g.638.11		24
21.20	even	2		105.2.x.a.53.11	yes	48
35.2	odd	12	inner	735.2.y.i.422.11		48
35.12	even	12		105.2.x.a.2.11	yes	48
35.13	even	4		525.2.bf.f.32.11		48
35.17	even	12		735.2.j.g.197.11		24
35.19	odd	6		525.2.bf.f.443.2		48
35.27	even	4		105.2.x.a.32.2	yes	48
35.32	odd	12		735.2.j.e.197.11		24
35.33	even	12		525.2.bf.f.107.2		48
35.34	odd	2		525.2.bf.f.368.11		48
105.2	even	12	inner	735.2.y.i.422.2		48
105.17	odd	12		735.2.j.g.197.2		24
105.32	even	12		735.2.j.e.197.2		24
105.47	odd	12		105.2.x.a.2.2	✓	48
105.62	odd	4		105.2.x.a.32.11	yes	48
105.68	odd	12		525.2.bf.f.107.11		48
105.83	odd	4		525.2.bf.f.32.2		48
105.89	even	6		525.2.bf.f.443.11		48
105.104	even	2		525.2.bf.f.368.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.2	✓	48	105.47	odd	12
105.2.x.a.2.11	yes	48	35.12	even	12
105.2.x.a.23.2	yes	48	21.5	even	6
105.2.x.a.23.11	yes	48	7.5	odd	6
105.2.x.a.32.2	yes	48	35.27	even	4
105.2.x.a.32.11	yes	48	105.62	odd	4
105.2.x.a.53.2	yes	48	7.6	odd	2
105.2.x.a.53.11	yes	48	21.20	even	2
525.2.bf.f.32.2		48	105.83	odd	4
525.2.bf.f.32.11		48	35.13	even	4
525.2.bf.f.107.2		48	35.33	even	12
525.2.bf.f.107.11		48	105.68	odd	12
525.2.bf.f.368.2		48	105.104	even	2
525.2.bf.f.368.11		48	35.34	odd	2
525.2.bf.f.443.2		48	35.19	odd	6
525.2.bf.f.443.11		48	105.89	even	6
735.2.j.e.197.2		24	105.32	even	12
735.2.j.e.197.11		24	35.32	odd	12
735.2.j.e.638.2		24	7.4	even	3
735.2.j.e.638.11		24	21.11	odd	6
735.2.j.g.197.2		24	105.17	odd	12
735.2.j.g.197.11		24	35.17	even	12
735.2.j.g.638.2		24	7.3	odd	6
735.2.j.g.638.11		24	21.17	even	6
735.2.y.i.128.2		48	21.2	odd	6	inner
735.2.y.i.128.11		48	7.2	even	3	inner
735.2.y.i.263.2		48	1.1	even	1	trivial
735.2.y.i.263.11		48	3.2	odd	2	inner
735.2.y.i.422.2		48	105.2	even	12	inner
735.2.y.i.422.11		48	35.2	odd	12	inner
735.2.y.i.557.2		48	5.2	odd	4	inner
735.2.y.i.557.11		48	15.2	even	4	inner