Embedded newform 572.2.bf.a.67.10

• Label: 572.2.bf.a.67.10
• Level: $572$
• Weight: $2$
• Character: 572.67
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.10
Character	\(\chi\)	\(=\)	572.67
Dual form			572.2.bf.a.111.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28912 + 0.581515i) q^{2} +(-0.101031 + 0.0583303i) q^{3} +(1.32368 - 1.49929i) q^{4} +(1.51781 - 1.51781i) q^{5} +(0.0963216 - 0.133946i) q^{6} +(-0.585492 - 2.18508i) q^{7} +(-0.834531 + 2.70251i) q^{8} +(-1.49320 + 2.58629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28912	+	0.581515i	−0.911548	+	0.411193i
							\(3\)	−0.101031	+	0.0583303i	−0.0583303	+	0.0336770i	−0.528882	−	0.848696i	\(-0.677388\pi\)
0.470551	+	0.882373i	\(0.344055\pi\)
\(5\)	1.51781	−	1.51781i	0.678785	−	0.678785i	−0.280940	−	0.959725i	\(-0.590646\pi\)
							0.959725	+	0.280940i	\(0.0906463\pi\)
\(7\)	−0.585492	−	2.18508i	−0.221295	−	0.825884i	−0.983855	−	0.178967i	\(-0.942725\pi\)
							0.762560	−	0.646917i	\(-0.223942\pi\)
\(11\)	0.965926	+	0.258819i	0.291238	+	0.0780369i
							\(13\)	3.24298	+	1.57579i	0.899440	+	0.437045i
\(17\)	2.91496	+	1.68295i	0.706981	+	0.408176i								0.809942	−	0.586510i	\(-0.199498\pi\)
							−0.102961	+	0.994685i	\(0.532832\pi\)
\(19\)	4.99073	−	1.33726i	1.14495	−	0.306789i	0.364012	−	0.931394i	\(-0.381407\pi\)
							0.780941	+	0.624605i	\(0.214740\pi\)
\(23\)	−4.67212	−	8.09235i	−0.974204	−	1.68737i	−0.682538	−	0.730850i	\(-0.739124\pi\)
							−0.291666	−	0.956520i	\(-0.594210\pi\)
\(29\)	−2.20767	−	3.82380i	−0.409955	−	0.710062i	0.584930	−	0.811084i	\(-0.301122\pi\)
							−0.994884	+	0.101022i	\(0.967789\pi\)
\(31\)	−0.269433	−	0.269433i	−0.0483915	−	0.0483915i	0.682497	−	0.730888i	\(-0.260894\pi\)
							−0.730888	+	0.682497i	\(0.760894\pi\)
\(37\)	2.96675	−	11.0721i	0.487731	−	1.82024i	−0.0797060	−	0.996818i	\(-0.525398\pi\)
							0.567437	−	0.823417i	\(-0.307935\pi\)
\(41\)	7.88214	+	2.11201i	1.23098	+	0.329841i	0.814962	−	0.579514i	\(-0.196758\pi\)
							0.416021	+	0.909355i	\(0.363424\pi\)
\(43\)	3.58355	−	6.20688i	0.546486	−	0.946541i	−0.452026	−	0.892005i	\(-0.649299\pi\)
							0.998512	−	0.0545361i	\(-0.0173680\pi\)
\(47\)	−2.46352	+	2.46352i	−0.359341	+	0.359341i	−0.863570	−	0.504229i	\(-0.831777\pi\)
							0.504229	+	0.863570i	\(0.331777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bf.a.67.10	✓	280
4.3	odd	2	inner	572.2.bf.a.67.22	yes	280
13.7	odd	12	inner	572.2.bf.a.111.22	yes	280
52.7	even	12	inner	572.2.bf.a.111.10	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bf.a.67.10	✓	280	1.1	even	1	trivial
572.2.bf.a.67.22	yes	280	4.3	odd	2	inner
572.2.bf.a.111.10	yes	280	52.7	even	12	inner
572.2.bf.a.111.22	yes	280	13.7	odd	12	inner