Embedded newform 735.2.j.e.638.6

• Label: 735.2.j.e.638.6
• Level: $735$
• Weight: $2$
• Character: 735.638
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $24$
• 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			638.6
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.e.197.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.218381 - 0.218381i) q^{2} +(-0.354425 + 1.69540i) q^{3} -1.90462i q^{4} +(-2.16448 - 0.561256i) q^{5} +(0.447643 - 0.292843i) q^{6} +(-0.852694 + 0.852694i) q^{8} +(-2.74877 - 1.20179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{3} + 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.218381	−	0.218381i	−0.154419	−	0.154419i	0.625670	−	0.780088i	\(-0.284826\pi\)
							−0.780088	+	0.625670i	\(0.784826\pi\)
\(3\)	−0.354425	+	1.69540i	−0.204628	+	0.978840i
							\(5\)	−2.16448	−	0.561256i	−0.967987	−	0.251001i
\(7\)		0			0
							\(11\)			0.762466i			0.229892i	0.993372	+	0.114946i	\(0.0366695\pi\)
−0.993372	+	0.114946i	\(0.963330\pi\)
\(13\)	2.27077	+	2.27077i	0.629797	+	0.629797i	0.948017	−	0.318220i	\(-0.103085\pi\)
							−0.318220	+	0.948017i	\(0.603085\pi\)
\(17\)	3.43677	+	3.43677i	0.833539	+	0.833539i	0.987999	−	0.154460i	\(-0.0493638\pi\)
							−0.154460	+	0.987999i	\(0.549364\pi\)
\(19\)			1.63691i			0.375533i	0.982214	+	0.187767i	\(0.0601248\pi\)
							−0.982214	+	0.187767i	\(0.939875\pi\)
\(23\)	5.40988	−	5.40988i	1.12804	−	1.12804i	0.137542	−	0.990496i	\(-0.456080\pi\)
							0.990496	−	0.137542i	\(-0.0439201\pi\)
\(29\)	4.94251			0.917801			0.458900	−	0.888488i	\(-0.348243\pi\)
							0.458900	+	0.888488i	\(0.348243\pi\)
\(31\)	5.92827			1.06475			0.532374	−	0.846509i	\(-0.321300\pi\)
							0.532374	+	0.846509i	\(0.321300\pi\)
\(37\)	−2.50059	+	2.50059i	−0.411095	+	0.411095i	−0.882120	−	0.471025i	\(-0.843884\pi\)
							0.471025	+	0.882120i	\(0.343884\pi\)
\(41\)			4.35963i			0.680860i	0.940270	+	0.340430i	\(0.110573\pi\)
							−0.940270	+	0.340430i	\(0.889427\pi\)
\(43\)	2.69037	+	2.69037i	0.410277	+	0.410277i	0.881835	−	0.471558i	\(-0.156308\pi\)
							−0.471558	+	0.881835i	\(0.656308\pi\)
\(47\)	3.03177	+	3.03177i	0.442230	+	0.442230i	0.892761	−	0.450531i	\(-0.148765\pi\)
							−0.450531	+	0.892761i	\(0.648765\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.e.638.6		24
3.2	odd	2	inner	735.2.j.e.638.7		24
5.2	odd	4	inner	735.2.j.e.197.7		24
7.2	even	3		735.2.y.i.263.6		48
7.3	odd	6		105.2.x.a.23.7	yes	48
7.4	even	3		735.2.y.i.128.7		48
7.5	odd	6		105.2.x.a.53.6	yes	48
7.6	odd	2		735.2.j.g.638.6		24
15.2	even	4	inner	735.2.j.e.197.6		24
21.2	odd	6		735.2.y.i.263.7		48
21.5	even	6		105.2.x.a.53.7	yes	48
21.11	odd	6		735.2.y.i.128.6		48
21.17	even	6		105.2.x.a.23.6	yes	48
21.20	even	2		735.2.j.g.638.7		24
35.2	odd	12		735.2.y.i.557.6		48
35.3	even	12		525.2.bf.f.107.6		48
35.12	even	12		105.2.x.a.32.6	yes	48
35.17	even	12		105.2.x.a.2.7	yes	48
35.19	odd	6		525.2.bf.f.368.7		48
35.24	odd	6		525.2.bf.f.443.6		48
35.27	even	4		735.2.j.g.197.7		24
35.32	odd	12		735.2.y.i.422.7		48
35.33	even	12		525.2.bf.f.32.7		48
105.2	even	12		735.2.y.i.557.7		48
105.17	odd	12		105.2.x.a.2.6	✓	48
105.32	even	12		735.2.y.i.422.6		48
105.38	odd	12		525.2.bf.f.107.7		48
105.47	odd	12		105.2.x.a.32.7	yes	48
105.59	even	6		525.2.bf.f.443.7		48
105.62	odd	4		735.2.j.g.197.6		24
105.68	odd	12		525.2.bf.f.32.6		48
105.89	even	6		525.2.bf.f.368.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.6	✓	48	105.17	odd	12
105.2.x.a.2.7	yes	48	35.17	even	12
105.2.x.a.23.6	yes	48	21.17	even	6
105.2.x.a.23.7	yes	48	7.3	odd	6
105.2.x.a.32.6	yes	48	35.12	even	12
105.2.x.a.32.7	yes	48	105.47	odd	12
105.2.x.a.53.6	yes	48	7.5	odd	6
105.2.x.a.53.7	yes	48	21.5	even	6
525.2.bf.f.32.6		48	105.68	odd	12
525.2.bf.f.32.7		48	35.33	even	12
525.2.bf.f.107.6		48	35.3	even	12
525.2.bf.f.107.7		48	105.38	odd	12
525.2.bf.f.368.6		48	105.89	even	6
525.2.bf.f.368.7		48	35.19	odd	6
525.2.bf.f.443.6		48	35.24	odd	6
525.2.bf.f.443.7		48	105.59	even	6
735.2.j.e.197.6		24	15.2	even	4	inner
735.2.j.e.197.7		24	5.2	odd	4	inner
735.2.j.e.638.6		24	1.1	even	1	trivial
735.2.j.e.638.7		24	3.2	odd	2	inner
735.2.j.g.197.6		24	105.62	odd	4
735.2.j.g.197.7		24	35.27	even	4
735.2.j.g.638.6		24	7.6	odd	2
735.2.j.g.638.7		24	21.20	even	2
735.2.y.i.128.6		48	21.11	odd	6
735.2.y.i.128.7		48	7.4	even	3
735.2.y.i.263.6		48	7.2	even	3
735.2.y.i.263.7		48	21.2	odd	6
735.2.y.i.422.6		48	105.32	even	12
735.2.y.i.422.7		48	35.32	odd	12
735.2.y.i.557.6		48	35.2	odd	12
735.2.y.i.557.7		48	105.2	even	12