Embedded newform 735.2.j.g.638.5

• Label: 735.2.j.g.638.5
• 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			638.5
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.g.197.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.664190 - 0.664190i) q^{2} +(0.578521 + 1.63258i) q^{3} -1.11770i q^{4} +(-0.459812 - 2.18828i) q^{5} +(0.700094 - 1.46859i) q^{6} +(-2.07075 + 2.07075i) q^{8} +(-2.33063 + 1.88896i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.664190	−	0.664190i	−0.469653	−	0.469653i	0.432149	−	0.901802i	\(-0.357755\pi\)
							−0.901802	+	0.432149i	\(0.857755\pi\)
\(3\)	0.578521	+	1.63258i	0.334010	+	0.942570i
							\(5\)	−0.459812	−	2.18828i	−0.205634	−	0.978629i
\(7\)		0			0
							\(11\)			0.727602i			0.219380i	0.993966	+	0.109690i	\(0.0349859\pi\)
−0.993966	+	0.109690i	\(0.965014\pi\)
\(13\)	−1.44243	−	1.44243i	−0.400058	−	0.400058i	0.478196	−	0.878253i	\(-0.341291\pi\)
							−0.878253	+	0.478196i	\(0.841291\pi\)
\(17\)	−5.19101	−	5.19101i	−1.25900	−	1.25900i	−0.951569	−	0.307435i	\(-0.900529\pi\)
							−0.307435	−	0.951569i	\(-0.599471\pi\)
\(19\)		−	0.767153i		−	0.175997i	−0.996121	−	0.0879985i	\(-0.971953\pi\)
							0.996121	−	0.0879985i	\(-0.0280471\pi\)
\(23\)	−2.29559	+	2.29559i	−0.478664	+	0.478664i	−0.904704	−	0.426040i	\(-0.859908\pi\)
							0.426040	+	0.904704i	\(0.359908\pi\)
\(29\)	4.07354			0.756437			0.378219	−	0.925716i	\(-0.376537\pi\)
							0.378219	+	0.925716i	\(0.376537\pi\)
\(31\)	0.419859			0.0754089			0.0377045	−	0.999289i	\(-0.487995\pi\)
							0.0377045	+	0.999289i	\(0.487995\pi\)
\(37\)	−4.45460	+	4.45460i	−0.732332	+	0.732332i	−0.971081	−	0.238749i	\(-0.923263\pi\)
							0.238749	+	0.971081i	\(0.423263\pi\)
\(41\)			4.44452i			0.694117i	0.937843	+	0.347058i	\(0.112819\pi\)
							−0.937843	+	0.347058i	\(0.887181\pi\)
\(43\)	−5.15881	−	5.15881i	−0.786711	−	0.786711i	0.194243	−	0.980953i	\(-0.437775\pi\)
							−0.980953	+	0.194243i	\(0.937775\pi\)
\(47\)	−4.97294	−	4.97294i	−0.725378	−	0.725378i	0.244317	−	0.969695i	\(-0.421436\pi\)
							−0.969695	+	0.244317i	\(0.921436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.g.638.5		24
3.2	odd	2	inner	735.2.j.g.638.8		24
5.2	odd	4	inner	735.2.j.g.197.8		24
7.2	even	3		105.2.x.a.53.5	yes	48
7.3	odd	6		735.2.y.i.128.8		48
7.4	even	3		105.2.x.a.23.8	yes	48
7.5	odd	6		735.2.y.i.263.5		48
7.6	odd	2		735.2.j.e.638.5		24
15.2	even	4	inner	735.2.j.g.197.5		24
21.2	odd	6		105.2.x.a.53.8	yes	48
21.5	even	6		735.2.y.i.263.8		48
21.11	odd	6		105.2.x.a.23.5	yes	48
21.17	even	6		735.2.y.i.128.5		48
21.20	even	2		735.2.j.e.638.8		24
35.2	odd	12		105.2.x.a.32.5	yes	48
35.4	even	6		525.2.bf.f.443.5		48
35.9	even	6		525.2.bf.f.368.8		48
35.12	even	12		735.2.y.i.557.5		48
35.17	even	12		735.2.y.i.422.8		48
35.18	odd	12		525.2.bf.f.107.5		48
35.23	odd	12		525.2.bf.f.32.8		48
35.27	even	4		735.2.j.e.197.8		24
35.32	odd	12		105.2.x.a.2.8	yes	48
105.2	even	12		105.2.x.a.32.8	yes	48
105.17	odd	12		735.2.y.i.422.5		48
105.23	even	12		525.2.bf.f.32.5		48
105.32	even	12		105.2.x.a.2.5	✓	48
105.44	odd	6		525.2.bf.f.368.5		48
105.47	odd	12		735.2.y.i.557.8		48
105.53	even	12		525.2.bf.f.107.8		48
105.62	odd	4		735.2.j.e.197.5		24
105.74	odd	6		525.2.bf.f.443.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.5	✓	48	105.32	even	12
105.2.x.a.2.8	yes	48	35.32	odd	12
105.2.x.a.23.5	yes	48	21.11	odd	6
105.2.x.a.23.8	yes	48	7.4	even	3
105.2.x.a.32.5	yes	48	35.2	odd	12
105.2.x.a.32.8	yes	48	105.2	even	12
105.2.x.a.53.5	yes	48	7.2	even	3
105.2.x.a.53.8	yes	48	21.2	odd	6
525.2.bf.f.32.5		48	105.23	even	12
525.2.bf.f.32.8		48	35.23	odd	12
525.2.bf.f.107.5		48	35.18	odd	12
525.2.bf.f.107.8		48	105.53	even	12
525.2.bf.f.368.5		48	105.44	odd	6
525.2.bf.f.368.8		48	35.9	even	6
525.2.bf.f.443.5		48	35.4	even	6
525.2.bf.f.443.8		48	105.74	odd	6
735.2.j.e.197.5		24	105.62	odd	4
735.2.j.e.197.8		24	35.27	even	4
735.2.j.e.638.5		24	7.6	odd	2
735.2.j.e.638.8		24	21.20	even	2
735.2.j.g.197.5		24	15.2	even	4	inner
735.2.j.g.197.8		24	5.2	odd	4	inner
735.2.j.g.638.5		24	1.1	even	1	trivial
735.2.j.g.638.8		24	3.2	odd	2	inner
735.2.y.i.128.5		48	21.17	even	6
735.2.y.i.128.8		48	7.3	odd	6
735.2.y.i.263.5		48	7.5	odd	6
735.2.y.i.263.8		48	21.5	even	6
735.2.y.i.422.5		48	105.17	odd	12
735.2.y.i.422.8		48	35.17	even	12
735.2.y.i.557.5		48	35.12	even	12
735.2.y.i.557.8		48	105.47	odd	12