Embedded newform 735.2.j.h.197.5

• Label: 735.2.j.h.197.5
• Level: $735$
• Weight: $2$
• Character: 735.197
• 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			197.5
Character	\(\chi\)	\(=\)	735.197
Dual form			735.2.j.h.638.5

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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	\(-0.805516\pi\)
							0.573677	+	0.819081i	\(0.305516\pi\)
\(3\)	−1.72305	+	0.176396i	−0.994801	+	0.101842i
							\(5\)	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	−0.939285	−	0.343137i	\(-0.888511\pi\)
							0.343137	+	0.939285i	\(0.388511\pi\)
\(17\)	−3.26719	+	3.26719i	−0.792410	+	0.792410i	−0.981886	−	0.189475i	\(-0.939321\pi\)
							0.189475	+	0.981886i	\(0.439321\pi\)
\(19\)		−	5.24329i		−	1.20289i	−0.798913	−	0.601446i	\(-0.794591\pi\)
							0.798913	−	0.601446i	\(-0.205409\pi\)
\(23\)	−2.54815	−	2.54815i	−0.531326	−	0.531326i	0.389641	−	0.920967i	\(-0.372599\pi\)
							−0.920967	+	0.389641i	\(0.872599\pi\)
\(29\)	−2.86924			−0.532804			−0.266402	−	0.963862i	\(-0.585835\pi\)
							−0.266402	+	0.963862i	\(0.585835\pi\)
\(31\)	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	0.508675	−	0.860959i	\(-0.330136\pi\)
							−0.860959	+	0.508675i	\(0.830136\pi\)
\(41\)		−	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	−0.646852	−	0.762615i	\(-0.723915\pi\)
							0.762615	+	0.646852i	\(0.223915\pi\)
\(47\)	7.66034	−	7.66034i	1.11738	−	1.11738i	0.125250	−	0.992125i	\(-0.460027\pi\)
							0.992125	−	0.125250i	\(-0.0399734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.h.197.5		24
3.2	odd	2	inner	735.2.j.h.197.8		24
5.3	odd	4	inner	735.2.j.h.638.8		24
7.2	even	3		735.2.y.g.557.8		48
7.3	odd	6		735.2.y.j.422.5		48
7.4	even	3		735.2.y.g.422.5		48
7.5	odd	6		735.2.y.j.557.8		48
7.6	odd	2		105.2.j.a.92.5	yes	24
15.8	even	4	inner	735.2.j.h.638.5		24
21.2	odd	6		735.2.y.g.557.5		48
21.5	even	6		735.2.y.j.557.5		48
21.11	odd	6		735.2.y.g.422.8		48
21.17	even	6		735.2.y.j.422.8		48
21.20	even	2		105.2.j.a.92.8	yes	24
35.3	even	12		735.2.y.j.128.5		48
35.13	even	4		105.2.j.a.8.8	yes	24
35.18	odd	12		735.2.y.g.128.5		48
35.23	odd	12		735.2.y.g.263.8		48
35.27	even	4		525.2.j.b.218.5		24
35.33	even	12		735.2.y.j.263.8		48
35.34	odd	2		525.2.j.b.407.8		24
105.23	even	12		735.2.y.g.263.5		48
105.38	odd	12		735.2.y.j.128.8		48
105.53	even	12		735.2.y.g.128.8		48
105.62	odd	4		525.2.j.b.218.8		24
105.68	odd	12		735.2.y.j.263.5		48
105.83	odd	4		105.2.j.a.8.5	✓	24
105.104	even	2		525.2.j.b.407.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.5	✓	24	105.83	odd	4
105.2.j.a.8.8	yes	24	35.13	even	4
105.2.j.a.92.5	yes	24	7.6	odd	2
105.2.j.a.92.8	yes	24	21.20	even	2
525.2.j.b.218.5		24	35.27	even	4
525.2.j.b.218.8		24	105.62	odd	4
525.2.j.b.407.5		24	105.104	even	2
525.2.j.b.407.8		24	35.34	odd	2
735.2.j.h.197.5		24	1.1	even	1	trivial
735.2.j.h.197.8		24	3.2	odd	2	inner
735.2.j.h.638.5		24	15.8	even	4	inner
735.2.j.h.638.8		24	5.3	odd	4	inner
735.2.y.g.128.5		48	35.18	odd	12
735.2.y.g.128.8		48	105.53	even	12
735.2.y.g.263.5		48	105.23	even	12
735.2.y.g.263.8		48	35.23	odd	12
735.2.y.g.422.5		48	7.4	even	3
735.2.y.g.422.8		48	21.11	odd	6
735.2.y.g.557.5		48	21.2	odd	6
735.2.y.g.557.8		48	7.2	even	3
735.2.y.j.128.5		48	35.3	even	12
735.2.y.j.128.8		48	105.38	odd	12
735.2.y.j.263.5		48	105.68	odd	12
735.2.y.j.263.8		48	35.33	even	12
735.2.y.j.422.5		48	7.3	odd	6
735.2.y.j.422.8		48	21.17	even	6
735.2.y.j.557.5		48	21.5	even	6
735.2.y.j.557.8		48	7.5	odd	6