Embedded newform 735.2.j.g.638.4

• Label: 735.2.j.g.638.4
• Level: $735$
• Weight: $2$
• Character: 735.638
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			638.4
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.g.197.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.929340 - 0.929340i) q^{2} +(-1.05286 + 1.37531i) q^{3} -0.272655i q^{4} +(0.980304 + 2.00973i) q^{5} +(2.25660 - 0.299670i) q^{6} +(-2.11207 + 2.11207i) q^{8} +(-0.782976 - 2.89602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} - 12 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.929340	−	0.929340i	−0.657142	−	0.657142i	0.297561	−	0.954703i	\(-0.403827\pi\)
							−0.954703	+	0.297561i	\(0.903827\pi\)
\(3\)	−1.05286	+	1.37531i	−0.607868	+	0.794038i
							\(5\)	0.980304	+	2.00973i	0.438405	+	0.898777i
\(7\)		0			0
							\(11\)		−	3.90548i		−	1.17755i	−0.808299	−	0.588773i	\(-0.799611\pi\)
0.808299	−	0.588773i	\(-0.200389\pi\)
\(13\)	−1.56642	−	1.56642i	−0.434448	−	0.434448i	0.455691	−	0.890138i	\(-0.349392\pi\)
							−0.890138	+	0.455691i	\(0.849392\pi\)
\(17\)	1.89349	+	1.89349i	0.459239	+	0.459239i	0.898406	−	0.439167i	\(-0.144726\pi\)
							−0.439167	+	0.898406i	\(0.644726\pi\)
\(19\)		−	1.86019i		−	0.426757i	−0.976970	−	0.213379i	\(-0.931553\pi\)
							0.976970	−	0.213379i	\(-0.0684468\pi\)
\(23\)	1.74459	−	1.74459i	0.363772	−	0.363772i	−0.501428	−	0.865200i	\(-0.667192\pi\)
							0.865200	+	0.501428i	\(0.167192\pi\)
\(29\)	0.513153			0.0952901			0.0476450	−	0.998864i	\(-0.484828\pi\)
							0.0476450	+	0.998864i	\(0.484828\pi\)
\(31\)	8.58277			1.54151			0.770755	−	0.637131i	\(-0.219879\pi\)
							0.770755	+	0.637131i	\(0.219879\pi\)
\(37\)	4.83665	−	4.83665i	0.795140	−	0.795140i	−0.187185	−	0.982325i	\(-0.559936\pi\)
							0.982325	+	0.187185i	\(0.0599363\pi\)
\(41\)		−	0.308469i		−	0.0481748i	−0.999710	−	0.0240874i	\(-0.992332\pi\)
							0.999710	−	0.0240874i	\(-0.00766800\pi\)
\(43\)	7.60892	+	7.60892i	1.16035	+	1.16035i	0.984399	+	0.175950i	\(0.0562999\pi\)
							0.175950	+	0.984399i	\(0.443700\pi\)
\(47\)	−3.74074	−	3.74074i	−0.545642	−	0.545642i	0.379535	−	0.925177i	\(-0.376084\pi\)
							−0.925177	+	0.379535i	\(0.876084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.g.638.4		24
3.2	odd	2	inner	735.2.j.g.638.9		24
5.2	odd	4	inner	735.2.j.g.197.9		24
7.2	even	3		105.2.x.a.53.4	yes	48
7.3	odd	6		735.2.y.i.128.9		48
7.4	even	3		105.2.x.a.23.9	yes	48
7.5	odd	6		735.2.y.i.263.4		48
7.6	odd	2		735.2.j.e.638.4		24
15.2	even	4	inner	735.2.j.g.197.4		24
21.2	odd	6		105.2.x.a.53.9	yes	48
21.5	even	6		735.2.y.i.263.9		48
21.11	odd	6		105.2.x.a.23.4	yes	48
21.17	even	6		735.2.y.i.128.4		48
21.20	even	2		735.2.j.e.638.9		24
35.2	odd	12		105.2.x.a.32.4	yes	48
35.4	even	6		525.2.bf.f.443.4		48
35.9	even	6		525.2.bf.f.368.9		48
35.12	even	12		735.2.y.i.557.4		48
35.17	even	12		735.2.y.i.422.9		48
35.18	odd	12		525.2.bf.f.107.4		48
35.23	odd	12		525.2.bf.f.32.9		48
35.27	even	4		735.2.j.e.197.9		24
35.32	odd	12		105.2.x.a.2.9	yes	48
105.2	even	12		105.2.x.a.32.9	yes	48
105.17	odd	12		735.2.y.i.422.4		48
105.23	even	12		525.2.bf.f.32.4		48
105.32	even	12		105.2.x.a.2.4	✓	48
105.44	odd	6		525.2.bf.f.368.4		48
105.47	odd	12		735.2.y.i.557.9		48
105.53	even	12		525.2.bf.f.107.9		48
105.62	odd	4		735.2.j.e.197.4		24
105.74	odd	6		525.2.bf.f.443.9		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.4	✓	48	105.32	even	12
105.2.x.a.2.9	yes	48	35.32	odd	12
105.2.x.a.23.4	yes	48	21.11	odd	6
105.2.x.a.23.9	yes	48	7.4	even	3
105.2.x.a.32.4	yes	48	35.2	odd	12
105.2.x.a.32.9	yes	48	105.2	even	12
105.2.x.a.53.4	yes	48	7.2	even	3
105.2.x.a.53.9	yes	48	21.2	odd	6
525.2.bf.f.32.4		48	105.23	even	12
525.2.bf.f.32.9		48	35.23	odd	12
525.2.bf.f.107.4		48	35.18	odd	12
525.2.bf.f.107.9		48	105.53	even	12
525.2.bf.f.368.4		48	105.44	odd	6
525.2.bf.f.368.9		48	35.9	even	6
525.2.bf.f.443.4		48	35.4	even	6
525.2.bf.f.443.9		48	105.74	odd	6
735.2.j.e.197.4		24	105.62	odd	4
735.2.j.e.197.9		24	35.27	even	4
735.2.j.e.638.4		24	7.6	odd	2
735.2.j.e.638.9		24	21.20	even	2
735.2.j.g.197.4		24	15.2	even	4	inner
735.2.j.g.197.9		24	5.2	odd	4	inner
735.2.j.g.638.4		24	1.1	even	1	trivial
735.2.j.g.638.9		24	3.2	odd	2	inner
735.2.y.i.128.4		48	21.17	even	6
735.2.y.i.128.9		48	7.3	odd	6
735.2.y.i.263.4		48	7.5	odd	6
735.2.y.i.263.9		48	21.5	even	6
735.2.y.i.422.4		48	105.17	odd	12
735.2.y.i.422.9		48	35.17	even	12
735.2.y.i.557.4		48	35.12	even	12
735.2.y.i.557.9		48	105.47	odd	12