Embedded newform 735.2.j.h.638.8

• Label: 735.2.j.h.638.8
• 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			638.8
Character	\(\chi\)	\(=\)	735.638
Dual form			735.2.j.h.197.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347054 + 0.347054i) q^{2} +(-0.176396 - 1.72305i) q^{3} -1.75911i q^{4} +(-1.16790 - 1.90683i) q^{5} +(0.536770 - 0.659208i) q^{6} +(1.30461 - 1.30461i) q^{8} +(-2.93777 + 0.607876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347054	+	0.347054i	0.245404	+	0.245404i	0.819081	−	0.573677i	\(-0.194484\pi\)
							−0.573677	+	0.819081i	\(0.694484\pi\)
\(3\)	−0.176396	−	1.72305i	−0.101842	−	0.994801i
							\(5\)	−1.16790	−	1.90683i	−0.522302	−	0.852760i
\(7\)		0			0
							\(11\)		−	2.67137i		−	0.805448i	−0.915322	−	0.402724i	\(-0.868064\pi\)
0.915322	−	0.402724i	\(-0.131936\pi\)
\(13\)	−2.14945	−	2.14945i	−0.596149	−	0.596149i	0.343137	−	0.939285i	\(-0.388511\pi\)
							−0.939285	+	0.343137i	\(0.888511\pi\)
\(17\)	3.26719	+	3.26719i	0.792410	+	0.792410i	0.981886	−	0.189475i	\(-0.0606787\pi\)
							−0.189475	+	0.981886i	\(0.560679\pi\)
\(19\)			5.24329i			1.20289i	0.798913	+	0.601446i	\(0.205409\pi\)
							−0.798913	+	0.601446i	\(0.794591\pi\)
\(23\)	2.54815	−	2.54815i	0.531326	−	0.531326i	−0.389641	−	0.920967i	\(-0.627401\pi\)
							0.920967	+	0.389641i	\(0.127401\pi\)
\(29\)	2.86924			0.532804			0.266402	−	0.963862i	\(-0.414165\pi\)
							0.266402	+	0.963862i	\(0.414165\pi\)
\(31\)	5.28599			0.949391			0.474696	−	0.880150i	\(-0.342558\pi\)
							0.474696	+	0.880150i	\(0.342558\pi\)
\(37\)	−2.14286	+	2.14286i	−0.352284	+	0.352284i	−0.860959	−	0.508675i	\(-0.830136\pi\)
							0.508675	+	0.860959i	\(0.330136\pi\)
\(41\)		−	11.5768i		−	1.80800i	−0.427537	−	0.903998i	\(-0.640619\pi\)
							0.427537	−	0.903998i	\(-0.359381\pi\)
\(43\)	0.759108	+	0.759108i	0.115763	+	0.115763i	0.762615	−	0.646852i	\(-0.223915\pi\)
							−0.646852	+	0.762615i	\(0.723915\pi\)
\(47\)	−7.66034	−	7.66034i	−1.11738	−	1.11738i	−0.992125	−	0.125250i	\(-0.960027\pi\)
							−0.125250	−	0.992125i	\(-0.539973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.h.638.8		24
3.2	odd	2	inner	735.2.j.h.638.5		24
5.2	odd	4	inner	735.2.j.h.197.5		24
7.2	even	3		735.2.y.g.263.8		48
7.3	odd	6		735.2.y.j.128.5		48
7.4	even	3		735.2.y.g.128.5		48
7.5	odd	6		735.2.y.j.263.8		48
7.6	odd	2		105.2.j.a.8.8	yes	24
15.2	even	4	inner	735.2.j.h.197.8		24
21.2	odd	6		735.2.y.g.263.5		48
21.5	even	6		735.2.y.j.263.5		48
21.11	odd	6		735.2.y.g.128.8		48
21.17	even	6		735.2.y.j.128.8		48
21.20	even	2		105.2.j.a.8.5	✓	24
35.2	odd	12		735.2.y.g.557.8		48
35.12	even	12		735.2.y.j.557.8		48
35.13	even	4		525.2.j.b.407.8		24
35.17	even	12		735.2.y.j.422.5		48
35.27	even	4		105.2.j.a.92.5	yes	24
35.32	odd	12		735.2.y.g.422.5		48
35.34	odd	2		525.2.j.b.218.5		24
105.2	even	12		735.2.y.g.557.5		48
105.17	odd	12		735.2.y.j.422.8		48
105.32	even	12		735.2.y.g.422.8		48
105.47	odd	12		735.2.y.j.557.5		48
105.62	odd	4		105.2.j.a.92.8	yes	24
105.83	odd	4		525.2.j.b.407.5		24
105.104	even	2		525.2.j.b.218.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.5	✓	24	21.20	even	2
105.2.j.a.8.8	yes	24	7.6	odd	2
105.2.j.a.92.5	yes	24	35.27	even	4
105.2.j.a.92.8	yes	24	105.62	odd	4
525.2.j.b.218.5		24	35.34	odd	2
525.2.j.b.218.8		24	105.104	even	2
525.2.j.b.407.5		24	105.83	odd	4
525.2.j.b.407.8		24	35.13	even	4
735.2.j.h.197.5		24	5.2	odd	4	inner
735.2.j.h.197.8		24	15.2	even	4	inner
735.2.j.h.638.5		24	3.2	odd	2	inner
735.2.j.h.638.8		24	1.1	even	1	trivial
735.2.y.g.128.5		48	7.4	even	3
735.2.y.g.128.8		48	21.11	odd	6
735.2.y.g.263.5		48	21.2	odd	6
735.2.y.g.263.8		48	7.2	even	3
735.2.y.g.422.5		48	35.32	odd	12
735.2.y.g.422.8		48	105.32	even	12
735.2.y.g.557.5		48	105.2	even	12
735.2.y.g.557.8		48	35.2	odd	12
735.2.y.j.128.5		48	7.3	odd	6
735.2.y.j.128.8		48	21.17	even	6
735.2.y.j.263.5		48	21.5	even	6
735.2.y.j.263.8		48	7.5	odd	6
735.2.y.j.422.5		48	35.17	even	12
735.2.y.j.422.8		48	105.17	odd	12
735.2.y.j.557.5		48	105.47	odd	12
735.2.y.j.557.8		48	35.12	even	12