Embedded newform 735.2.j.h.638.2

• Label: 735.2.j.h.638.2
• 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			638.2
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.h.197.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54414 - 1.54414i) q^{2} +(0.00622252 + 1.73204i) q^{3} +2.76875i q^{4} +(-0.252500 + 2.22177i) q^{5} +(2.66491 - 2.68412i) q^{6} +(1.18705 - 1.18705i) q^{8} +(-2.99992 + 0.0215553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54414	−	1.54414i	−1.09187	−	1.09187i	−0.995329	−	0.0965442i	\(-0.969221\pi\)
							−0.0965442	−	0.995329i	\(-0.530779\pi\)
\(3\)	0.00622252	+	1.73204i	0.00359257	+	0.999994i
							\(5\)	−0.252500	+	2.22177i	−0.112921	+	0.993604i
\(7\)		0			0
							\(11\)			3.38507i			1.02064i	0.859986	+	0.510318i	\(0.170472\pi\)
−0.859986	+	0.510318i	\(0.829528\pi\)
\(13\)	0.206632	+	0.206632i	0.0573094	+	0.0573094i	0.735181	−	0.677871i	\(-0.237097\pi\)
							−0.677871	+	0.735181i	\(0.737097\pi\)
\(17\)	0.167409	+	0.167409i	0.0406026	+	0.0406026i	0.727117	−	0.686514i	\(-0.240860\pi\)
							−0.686514	+	0.727117i	\(0.740860\pi\)
\(19\)			5.31419i			1.21916i	0.792725	+	0.609579i	\(0.208662\pi\)
							−0.792725	+	0.609579i	\(0.791338\pi\)
\(23\)	5.07773	−	5.07773i	1.05878	−	1.05878i	0.0606179	−	0.998161i	\(-0.480693\pi\)
							0.998161	−	0.0606179i	\(-0.0193071\pi\)
\(29\)	2.84268			0.527872			0.263936	−	0.964540i	\(-0.414979\pi\)
							0.263936	+	0.964540i	\(0.414979\pi\)
\(31\)	−9.11776			−1.63760			−0.818799	−	0.574081i	\(-0.805360\pi\)
							−0.818799	+	0.574081i	\(0.805360\pi\)
\(37\)	−5.27013	+	5.27013i	−0.866404	+	0.866404i	−0.992072	−	0.125668i	\(-0.959893\pi\)
							0.125668	+	0.992072i	\(0.459893\pi\)
\(41\)		−	0.0314968i		−	0.00491898i	−0.999997	−	0.00245949i	\(-0.999217\pi\)
							0.999997	−	0.00245949i	\(-0.000782881\pi\)
\(43\)	−3.76875	−	3.76875i	−0.574728	−	0.574728i	0.358718	−	0.933446i	\(-0.383214\pi\)
							−0.933446	+	0.358718i	\(0.883214\pi\)
\(47\)	−3.56639	−	3.56639i	−0.520211	−	0.520211i	0.397424	−	0.917635i	\(-0.369904\pi\)
							−0.917635	+	0.397424i	\(0.869904\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.h.638.2		24
3.2	odd	2	inner	735.2.j.h.638.11		24
5.2	odd	4	inner	735.2.j.h.197.11		24
7.2	even	3		735.2.y.g.263.2		48
7.3	odd	6		735.2.y.j.128.11		48
7.4	even	3		735.2.y.g.128.11		48
7.5	odd	6		735.2.y.j.263.2		48
7.6	odd	2		105.2.j.a.8.2	✓	24
15.2	even	4	inner	735.2.j.h.197.2		24
21.2	odd	6		735.2.y.g.263.11		48
21.5	even	6		735.2.y.j.263.11		48
21.11	odd	6		735.2.y.g.128.2		48
21.17	even	6		735.2.y.j.128.2		48
21.20	even	2		105.2.j.a.8.11	yes	24
35.2	odd	12		735.2.y.g.557.2		48
35.12	even	12		735.2.y.j.557.2		48
35.13	even	4		525.2.j.b.407.2		24
35.17	even	12		735.2.y.j.422.11		48
35.27	even	4		105.2.j.a.92.11	yes	24
35.32	odd	12		735.2.y.g.422.11		48
35.34	odd	2		525.2.j.b.218.11		24
105.2	even	12		735.2.y.g.557.11		48
105.17	odd	12		735.2.y.j.422.2		48
105.32	even	12		735.2.y.g.422.2		48
105.47	odd	12		735.2.y.j.557.11		48
105.62	odd	4		105.2.j.a.92.2	yes	24
105.83	odd	4		525.2.j.b.407.11		24
105.104	even	2		525.2.j.b.218.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.2	✓	24	7.6	odd	2
105.2.j.a.8.11	yes	24	21.20	even	2
105.2.j.a.92.2	yes	24	105.62	odd	4
105.2.j.a.92.11	yes	24	35.27	even	4
525.2.j.b.218.2		24	105.104	even	2
525.2.j.b.218.11		24	35.34	odd	2
525.2.j.b.407.2		24	35.13	even	4
525.2.j.b.407.11		24	105.83	odd	4
735.2.j.h.197.2		24	15.2	even	4	inner
735.2.j.h.197.11		24	5.2	odd	4	inner
735.2.j.h.638.2		24	1.1	even	1	trivial
735.2.j.h.638.11		24	3.2	odd	2	inner
735.2.y.g.128.2		48	21.11	odd	6
735.2.y.g.128.11		48	7.4	even	3
735.2.y.g.263.2		48	7.2	even	3
735.2.y.g.263.11		48	21.2	odd	6
735.2.y.g.422.2		48	105.32	even	12
735.2.y.g.422.11		48	35.32	odd	12
735.2.y.g.557.2		48	35.2	odd	12
735.2.y.g.557.11		48	105.2	even	12
735.2.y.j.128.2		48	21.17	even	6
735.2.y.j.128.11		48	7.3	odd	6
735.2.y.j.263.2		48	7.5	odd	6
735.2.y.j.263.11		48	21.5	even	6
735.2.y.j.422.2		48	105.17	odd	12
735.2.y.j.422.11		48	35.17	even	12
735.2.y.j.557.2		48	35.12	even	12
735.2.y.j.557.11		48	105.47	odd	12