Embedded newform 735.2.j.g.638.11

• Label: 735.2.j.g.638.11
• 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			638.11
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.g.197.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.59037 + 1.59037i) q^{2} +(-1.06967 - 1.36228i) q^{3} +3.05858i q^{4} +(0.812581 - 2.08320i) q^{5} +(0.465359 - 3.86771i) q^{6} +(-1.68355 + 1.68355i) q^{8} +(-0.711613 + 2.91438i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.59037	+	1.59037i	1.12456	+	1.12456i	0.991046	+	0.133519i	\(0.0426276\pi\)
							0.133519	+	0.991046i	\(0.457372\pi\)
\(3\)	−1.06967	−	1.36228i	−0.617574	−	0.786513i
							\(5\)	0.812581	−	2.08320i	0.363397	−	0.931634i
\(7\)		0			0
							\(11\)		−	4.48865i		−	1.35338i	−0.736268	−	0.676690i	\(-0.763414\pi\)
0.736268	−	0.676690i	\(-0.236586\pi\)
\(13\)	−1.08424	−	1.08424i	−0.300715	−	0.300715i	0.540578	−	0.841294i	\(-0.318205\pi\)
							−0.841294	+	0.540578i	\(0.818205\pi\)
\(17\)	1.49970	+	1.49970i	0.363731	+	0.363731i	0.865185	−	0.501454i	\(-0.167201\pi\)
							−0.501454	+	0.865185i	\(0.667201\pi\)
\(19\)		−	4.22707i		−	0.969757i	−0.874582	−	0.484878i	\(-0.838864\pi\)
							0.874582	−	0.484878i	\(-0.161136\pi\)
\(23\)	2.29591	−	2.29591i	0.478731	−	0.478731i	−0.425995	−	0.904726i	\(-0.640076\pi\)
							0.904726	+	0.425995i	\(0.140076\pi\)
\(29\)	1.69118			0.314045			0.157023	−	0.987595i	\(-0.449810\pi\)
							0.157023	+	0.987595i	\(0.449810\pi\)
\(31\)	−1.06015			−0.190409			−0.0952047	−	0.995458i	\(-0.530351\pi\)
							−0.0952047	+	0.995458i	\(0.530351\pi\)
\(37\)	4.21494	−	4.21494i	0.692931	−	0.692931i	−0.269944	−	0.962876i	\(-0.587005\pi\)
							0.962876	+	0.269944i	\(0.0870053\pi\)
\(41\)			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	\(0.680678\pi\)
\(47\)	3.73368	+	3.73368i	0.544613	+	0.544613i	0.924878	−	0.380265i	\(-0.124167\pi\)
							−0.380265	+	0.924878i	\(0.624167\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.2	✓	48	35.32	odd	12
105.2.x.a.2.11	yes	48	105.32	even	12
105.2.x.a.23.2	yes	48	7.4	even	3
105.2.x.a.23.11	yes	48	21.11	odd	6
105.2.x.a.32.2	yes	48	105.2	even	12
105.2.x.a.32.11	yes	48	35.2	odd	12
105.2.x.a.53.2	yes	48	21.2	odd	6
105.2.x.a.53.11	yes	48	7.2	even	3
525.2.bf.f.32.2		48	35.23	odd	12
525.2.bf.f.32.11		48	105.23	even	12
525.2.bf.f.107.2		48	105.53	even	12
525.2.bf.f.107.11		48	35.18	odd	12
525.2.bf.f.368.2		48	35.9	even	6
525.2.bf.f.368.11		48	105.44	odd	6
525.2.bf.f.443.2		48	105.74	odd	6
525.2.bf.f.443.11		48	35.4	even	6
735.2.j.e.197.2		24	35.27	even	4
735.2.j.e.197.11		24	105.62	odd	4
735.2.j.e.638.2		24	21.20	even	2
735.2.j.e.638.11		24	7.6	odd	2
735.2.j.g.197.2		24	5.2	odd	4	inner
735.2.j.g.197.11		24	15.2	even	4	inner
735.2.j.g.638.2		24	3.2	odd	2	inner
735.2.j.g.638.11		24	1.1	even	1	trivial
735.2.y.i.128.2		48	7.3	odd	6
735.2.y.i.128.11		48	21.17	even	6
735.2.y.i.263.2		48	21.5	even	6
735.2.y.i.263.11		48	7.5	odd	6
735.2.y.i.422.2		48	35.17	even	12
735.2.y.i.422.11		48	105.17	odd	12
735.2.y.i.557.2		48	105.47	odd	12
735.2.y.i.557.11		48	35.12	even	12