Embedded newform 735.2.j.e.638.1

• Label: 735.2.j.e.638.1
• 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			638.1
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.e.197.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72500 - 1.72500i) q^{2} +(-1.44593 + 0.953569i) q^{3} +3.95128i q^{4} +(-1.96293 + 1.07094i) q^{5} +(4.13914 + 0.849321i) q^{6} +(3.36596 - 3.36596i) q^{8} +(1.18141 - 2.75758i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{3} + 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72500	−	1.72500i	−1.21976	−	1.21976i	−0.967715	−	0.252047i	\(-0.918896\pi\)
							−0.252047	−	0.967715i	\(-0.581104\pi\)
\(3\)	−1.44593	+	0.953569i	−0.834807	+	0.550543i
							\(5\)	−1.96293	+	1.07094i	−0.877848	+	0.478939i
\(7\)		0			0
							\(11\)		−	3.55709i		−	1.07250i	−0.844058	−	0.536252i	\(-0.819840\pi\)
0.844058	−	0.536252i	\(-0.180160\pi\)
\(13\)	−1.28412	−	1.28412i	−0.356151	−	0.356151i	0.506241	−	0.862392i	\(-0.331035\pi\)
							−0.862392	+	0.506241i	\(0.831035\pi\)
\(17\)	−2.16418	−	2.16418i	−0.524891	−	0.524891i	0.394154	−	0.919045i	\(-0.371038\pi\)
							−0.919045	+	0.394154i	\(0.871038\pi\)
\(19\)			0.383034i			0.0878740i	0.999034	+	0.0439370i	\(0.0139901\pi\)
							−0.999034	+	0.0439370i	\(0.986010\pi\)
\(23\)	−1.79948	+	1.79948i	−0.375218	+	0.375218i	−0.869374	−	0.494156i	\(-0.835477\pi\)
							0.494156	+	0.869374i	\(0.335477\pi\)
\(29\)	5.51741			1.02456			0.512279	−	0.858819i	\(-0.328801\pi\)
							0.512279	+	0.858819i	\(0.328801\pi\)
\(31\)	0.647960			0.116377			0.0581885	−	0.998306i	\(-0.481468\pi\)
							0.0581885	+	0.998306i	\(0.481468\pi\)
\(37\)	−3.66372	+	3.66372i	−0.602311	+	0.602311i	−0.940925	−	0.338614i	\(-0.890042\pi\)
							0.338614	+	0.940925i	\(0.390042\pi\)
\(41\)			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	\(0.761509\pi\)
\(47\)	2.05365	+	2.05365i	0.299555	+	0.299555i	0.840840	−	0.541284i	\(-0.182062\pi\)
							−0.541284	+	0.840840i	\(0.682062\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.1	✓	48	105.17	odd	12
105.2.x.a.2.12	yes	48	35.17	even	12
105.2.x.a.23.1	yes	48	21.17	even	6
105.2.x.a.23.12	yes	48	7.3	odd	6
105.2.x.a.32.1	yes	48	35.12	even	12
105.2.x.a.32.12	yes	48	105.47	odd	12
105.2.x.a.53.1	yes	48	7.5	odd	6
105.2.x.a.53.12	yes	48	21.5	even	6
525.2.bf.f.32.1		48	105.68	odd	12
525.2.bf.f.32.12		48	35.33	even	12
525.2.bf.f.107.1		48	35.3	even	12
525.2.bf.f.107.12		48	105.38	odd	12
525.2.bf.f.368.1		48	105.89	even	6
525.2.bf.f.368.12		48	35.19	odd	6
525.2.bf.f.443.1		48	35.24	odd	6
525.2.bf.f.443.12		48	105.59	even	6
735.2.j.e.197.1		24	15.2	even	4	inner
735.2.j.e.197.12		24	5.2	odd	4	inner
735.2.j.e.638.1		24	1.1	even	1	trivial
735.2.j.e.638.12		24	3.2	odd	2	inner
735.2.j.g.197.1		24	105.62	odd	4
735.2.j.g.197.12		24	35.27	even	4
735.2.j.g.638.1		24	7.6	odd	2
735.2.j.g.638.12		24	21.20	even	2
735.2.y.i.128.1		48	21.11	odd	6
735.2.y.i.128.12		48	7.4	even	3
735.2.y.i.263.1		48	7.2	even	3
735.2.y.i.263.12		48	21.2	odd	6
735.2.y.i.422.1		48	105.32	even	12
735.2.y.i.422.12		48	35.32	odd	12
735.2.y.i.557.1		48	35.2	odd	12
735.2.y.i.557.12		48	105.2	even	12