Embedded newform 735.2.y.j.128.10

• Label: 735.2.y.j.128.10
• Level: $735$
• Weight: $2$
• Character: 735.128
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			128.10
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.j.557.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.69953 - 0.455388i) q^{2} +(-0.421844 - 1.67990i) q^{3} +(0.948973 - 0.547890i) q^{4} +(-2.12027 + 0.710238i) q^{5} +(-1.48194 - 2.66293i) q^{6} +(-1.12498 + 1.12498i) q^{8} +(-2.64410 + 1.41731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.69953	−	0.455388i	1.20175	−	0.322008i	0.398229	−	0.917286i	\(-0.369625\pi\)
							0.803520	+	0.595278i	\(0.202958\pi\)
\(3\)	−0.421844	−	1.67990i	−0.243552	−	0.969888i
							\(5\)	−2.12027	+	0.710238i	−0.948215	+	0.317628i
\(7\)		0			0
							\(11\)	−1.34425	+	0.776103i	−0.405307	+	0.234004i	−0.688771	−	0.724979i	\(-0.741850\pi\)
0.283465	+	0.958983i	\(0.408516\pi\)
\(13\)	−4.50889	−	4.50889i	−1.25054	−	1.25054i	−0.955478	−	0.295062i	\(-0.904660\pi\)
							−0.295062	−	0.955478i	\(-0.595340\pi\)
\(17\)	−0.780989	+	2.91469i	−0.189418	+	0.706917i	0.804224	+	0.594327i	\(0.202581\pi\)
							−0.993642	+	0.112590i	\(0.964085\pi\)
\(19\)	−3.64590	−	2.10496i	−0.836427	−	0.482912i	0.0196209	−	0.999807i	\(-0.493754\pi\)
							−0.856048	+	0.516896i	\(0.827087\pi\)
\(23\)	−1.37644	−	5.13695i	−0.287008	−	1.07113i	−0.947360	−	0.320171i	\(-0.896260\pi\)
							0.660352	−	0.750956i	\(-0.270407\pi\)
\(29\)	2.97115			0.551728			0.275864	−	0.961197i	\(-0.411036\pi\)
							0.275864	+	0.961197i	\(0.411036\pi\)
\(31\)	2.89885	+	5.02095i	0.520649	+	0.901790i	0.999712	+	0.0240096i	\(0.00764322\pi\)
							−0.479063	+	0.877781i	\(0.659023\pi\)
\(37\)	−0.450663	−	1.68190i	−0.0740885	−	0.276502i	0.918937	−	0.394405i	\(-0.129049\pi\)
							−0.993025	+	0.117903i	\(0.962383\pi\)
\(41\)		−	2.68458i		−	0.419261i	−0.977781	−	0.209631i	\(-0.932774\pi\)
							0.977781	−	0.209631i	\(-0.0672261\pi\)
\(43\)	−2.09578	−	2.09578i	−0.319603	−	0.319603i	0.529011	−	0.848615i	\(-0.322563\pi\)
							−0.848615	+	0.529011i	\(0.822563\pi\)
\(47\)	0.0489622	−	0.0131194i	0.00714187	−	0.00191366i	−0.255246	−	0.966876i	\(-0.582157\pi\)
							0.262388	+	0.964962i	\(0.415490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.j.128.10		48
3.2	odd	2	inner	735.2.y.j.128.3		48
5.2	odd	4	inner	735.2.y.j.422.10		48
7.2	even	3		105.2.j.a.8.3	✓	24
7.3	odd	6		735.2.y.g.263.3		48
7.4	even	3	inner	735.2.y.j.263.3		48
7.5	odd	6		735.2.j.h.638.3		24
7.6	odd	2		735.2.y.g.128.10		48
15.2	even	4	inner	735.2.y.j.422.3		48
21.2	odd	6		105.2.j.a.8.10	yes	24
21.5	even	6		735.2.j.h.638.10		24
21.11	odd	6	inner	735.2.y.j.263.10		48
21.17	even	6		735.2.y.g.263.10		48
21.20	even	2		735.2.y.g.128.3		48
35.2	odd	12		105.2.j.a.92.10	yes	24
35.9	even	6		525.2.j.b.218.10		24
35.12	even	12		735.2.j.h.197.10		24
35.17	even	12		735.2.y.g.557.3		48
35.23	odd	12		525.2.j.b.407.3		24
35.27	even	4		735.2.y.g.422.10		48
35.32	odd	12	inner	735.2.y.j.557.3		48
105.2	even	12		105.2.j.a.92.3	yes	24
105.17	odd	12		735.2.y.g.557.10		48
105.23	even	12		525.2.j.b.407.10		24
105.32	even	12	inner	735.2.y.j.557.10		48
105.44	odd	6		525.2.j.b.218.3		24
105.47	odd	12		735.2.j.h.197.3		24
105.62	odd	4		735.2.y.g.422.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.3	✓	24	7.2	even	3
105.2.j.a.8.10	yes	24	21.2	odd	6
105.2.j.a.92.3	yes	24	105.2	even	12
105.2.j.a.92.10	yes	24	35.2	odd	12
525.2.j.b.218.3		24	105.44	odd	6
525.2.j.b.218.10		24	35.9	even	6
525.2.j.b.407.3		24	35.23	odd	12
525.2.j.b.407.10		24	105.23	even	12
735.2.j.h.197.3		24	105.47	odd	12
735.2.j.h.197.10		24	35.12	even	12
735.2.j.h.638.3		24	7.5	odd	6
735.2.j.h.638.10		24	21.5	even	6
735.2.y.g.128.3		48	21.20	even	2
735.2.y.g.128.10		48	7.6	odd	2
735.2.y.g.263.3		48	7.3	odd	6
735.2.y.g.263.10		48	21.17	even	6
735.2.y.g.422.3		48	105.62	odd	4
735.2.y.g.422.10		48	35.27	even	4
735.2.y.g.557.3		48	35.17	even	12
735.2.y.g.557.10		48	105.17	odd	12
735.2.y.j.128.3		48	3.2	odd	2	inner
735.2.y.j.128.10		48	1.1	even	1	trivial
735.2.y.j.263.3		48	7.4	even	3	inner
735.2.y.j.263.10		48	21.11	odd	6	inner
735.2.y.j.422.3		48	15.2	even	4	inner
735.2.y.j.422.10		48	5.2	odd	4	inner
735.2.y.j.557.3		48	35.32	odd	12	inner
735.2.y.j.557.10		48	105.32	even	12	inner