Embedded newform 735.2.j.h.197.10

• Label: 735.2.j.h.197.10
• 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			197.10
Character	\(\chi\)	\(=\)	735.197
Dual form			735.2.j.h.638.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24414 - 1.24414i) q^{2} +(-0.474620 + 1.66575i) q^{3} -1.09578i q^{4} +(1.67522 - 1.48109i) q^{5} +(1.48194 + 2.66293i) q^{6} +(1.12498 + 1.12498i) q^{8} +(-2.54947 - 1.58120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24414	−	1.24414i	0.879741	−	0.879741i	−0.113766	−	0.993508i	\(-0.536291\pi\)
							0.993508	+	0.113766i	\(0.0362914\pi\)
\(3\)	−0.474620	+	1.66575i	−0.274022	+	0.961723i
							\(5\)	1.67522	−	1.48109i	0.749182	−	0.662365i
\(7\)		0			0
							\(11\)		−	1.55221i		−	0.468008i	−0.972236	−	0.234004i	\(-0.924817\pi\)
0.972236	−	0.234004i	\(-0.0751828\pi\)
\(13\)	4.50889	−	4.50889i	1.25054	−	1.25054i	0.295062	−	0.955478i	\(-0.404660\pi\)
							0.955478	−	0.295062i	\(-0.0953402\pi\)
\(17\)	−2.13370	+	2.13370i	−0.517499	+	0.517499i	−0.916814	−	0.399315i	\(-0.869248\pi\)
							0.399315	+	0.916814i	\(0.369248\pi\)
\(19\)			4.20993i			0.965823i	0.875669	+	0.482912i	\(0.160421\pi\)
							−0.875669	+	0.482912i	\(0.839579\pi\)
\(23\)	3.76050	+	3.76050i	0.784119	+	0.784119i	0.980523	−	0.196404i	\(-0.0629263\pi\)
							−0.196404	+	0.980523i	\(0.562926\pi\)
\(29\)	−2.97115			−0.551728			−0.275864	−	0.961197i	\(-0.588964\pi\)
							−0.275864	+	0.961197i	\(0.588964\pi\)
\(31\)	5.79770			1.04130			0.520649	−	0.853771i	\(-0.325690\pi\)
							0.520649	+	0.853771i	\(0.325690\pi\)
\(37\)	−1.23123	−	1.23123i	−0.202414	−	0.202414i	0.598620	−	0.801033i	\(-0.295716\pi\)
							−0.801033	+	0.598620i	\(0.795716\pi\)
\(41\)			2.68458i			0.419261i	0.977781	+	0.209631i	\(0.0672261\pi\)
							−0.977781	+	0.209631i	\(0.932774\pi\)
\(43\)	−2.09578	+	2.09578i	−0.319603	+	0.319603i	−0.848615	−	0.529011i	\(-0.822563\pi\)
							0.529011	+	0.848615i	\(0.322563\pi\)
\(47\)	−0.0358428	+	0.0358428i	−0.00522821	+	0.00522821i	−0.709716	−	0.704488i	\(-0.751177\pi\)
							0.704488	+	0.709716i	\(0.251177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.j.h.197.10		24
3.2	odd	2	inner	735.2.j.h.197.3		24
5.3	odd	4	inner	735.2.j.h.638.3		24
7.2	even	3		735.2.y.g.557.3		48
7.3	odd	6		735.2.y.j.422.10		48
7.4	even	3		735.2.y.g.422.10		48
7.5	odd	6		735.2.y.j.557.3		48
7.6	odd	2		105.2.j.a.92.10	yes	24
15.8	even	4	inner	735.2.j.h.638.10		24
21.2	odd	6		735.2.y.g.557.10		48
21.5	even	6		735.2.y.j.557.10		48
21.11	odd	6		735.2.y.g.422.3		48
21.17	even	6		735.2.y.j.422.3		48
21.20	even	2		105.2.j.a.92.3	yes	24
35.3	even	12		735.2.y.j.128.10		48
35.13	even	4		105.2.j.a.8.3	✓	24
35.18	odd	12		735.2.y.g.128.10		48
35.23	odd	12		735.2.y.g.263.3		48
35.27	even	4		525.2.j.b.218.10		24
35.33	even	12		735.2.y.j.263.3		48
35.34	odd	2		525.2.j.b.407.3		24
105.23	even	12		735.2.y.g.263.10		48
105.38	odd	12		735.2.y.j.128.3		48
105.53	even	12		735.2.y.g.128.3		48
105.62	odd	4		525.2.j.b.218.3		24
105.68	odd	12		735.2.y.j.263.10		48
105.83	odd	4		105.2.j.a.8.10	yes	24
105.104	even	2		525.2.j.b.407.10		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.3	✓	24	35.13	even	4
105.2.j.a.8.10	yes	24	105.83	odd	4
105.2.j.a.92.3	yes	24	21.20	even	2
105.2.j.a.92.10	yes	24	7.6	odd	2
525.2.j.b.218.3		24	105.62	odd	4
525.2.j.b.218.10		24	35.27	even	4
525.2.j.b.407.3		24	35.34	odd	2
525.2.j.b.407.10		24	105.104	even	2
735.2.j.h.197.3		24	3.2	odd	2	inner
735.2.j.h.197.10		24	1.1	even	1	trivial
735.2.j.h.638.3		24	5.3	odd	4	inner
735.2.j.h.638.10		24	15.8	even	4	inner
735.2.y.g.128.3		48	105.53	even	12
735.2.y.g.128.10		48	35.18	odd	12
735.2.y.g.263.3		48	35.23	odd	12
735.2.y.g.263.10		48	105.23	even	12
735.2.y.g.422.3		48	21.11	odd	6
735.2.y.g.422.10		48	7.4	even	3
735.2.y.g.557.3		48	7.2	even	3
735.2.y.g.557.10		48	21.2	odd	6
735.2.y.j.128.3		48	105.38	odd	12
735.2.y.j.128.10		48	35.3	even	12
735.2.y.j.263.3		48	35.33	even	12
735.2.y.j.263.10		48	105.68	odd	12
735.2.y.j.422.3		48	21.17	even	6
735.2.y.j.422.10		48	7.3	odd	6
735.2.y.j.557.3		48	7.5	odd	6
735.2.y.j.557.10		48	21.5	even	6