Embedded newform 572.2.bf.a.67.17

• Label: 572.2.bf.a.67.17
• 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			67.17
Character	\(\chi\)	\(=\)	572.67
Dual form			572.2.bf.a.111.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07247 + 0.921853i) q^{2} +(0.790362 - 0.456316i) q^{3} +(0.300374 - 1.97732i) q^{4} +(0.301911 - 0.301911i) q^{5} +(-0.426981 + 1.21798i) q^{6} +(0.768929 + 2.86968i) q^{7} +(1.50065 + 2.39751i) q^{8} +(-1.08355 + 1.87677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07247	+	0.921853i	−0.758349	+	0.651849i
							\(3\)	0.790362	−	0.456316i	0.456316	−	0.263454i	−0.254178	−	0.967157i	\(-0.581805\pi\)
0.710494	+	0.703704i	\(0.248472\pi\)
\(5\)	0.301911	−	0.301911i	0.135019	−	0.135019i	−0.636367	−	0.771386i	\(-0.719564\pi\)
							0.771386	+	0.636367i	\(0.219564\pi\)
\(7\)	0.768929	+	2.86968i	0.290628	+	1.08464i	0.944628	+	0.328143i	\(0.106423\pi\)
							−0.654000	+	0.756494i	\(0.726911\pi\)
\(11\)	−0.965926	−	0.258819i	−0.291238	−	0.0780369i
							\(13\)	−1.31145	+	3.35859i	−0.363730	+	0.931504i
\(17\)	−2.17107	−	1.25347i	−0.526562	−	0.304011i								0.213053	−	0.977041i	\(-0.431659\pi\)
							−0.739615	+	0.673030i	\(0.764993\pi\)
\(19\)	4.31656	−	1.15662i	0.990288	−	0.265347i	0.272916	−	0.962038i	\(-0.412012\pi\)
							0.717371	+	0.696691i	\(0.245345\pi\)
\(23\)	−0.258660	−	0.448013i	−0.0539344	−	0.0934171i	0.837798	−	0.545981i	\(-0.183843\pi\)
							−0.891732	+	0.452564i	\(0.850510\pi\)
\(29\)	2.75689	+	4.77507i	0.511942	+	0.886709i	0.999904	+	0.0138444i	\(0.00440696\pi\)
							−0.487962	+	0.872865i	\(0.662260\pi\)
\(31\)	−3.58556	−	3.58556i	−0.643986	−	0.643986i	0.307547	−	0.951533i	\(-0.400492\pi\)
							−0.951533	+	0.307547i	\(0.900492\pi\)
\(37\)	−2.91309	+	10.8718i	−0.478909	+	1.78731i	0.127137	+	0.991885i	\(0.459421\pi\)
							−0.606047	+	0.795429i	\(0.707246\pi\)
\(41\)	11.3956	+	3.05344i	1.77969	+	0.476867i	0.990527	−	0.137316i	\(-0.0438475\pi\)
							0.789164	+	0.614183i	\(0.210514\pi\)
\(43\)	0.144005	−	0.249425i	0.0219606	−	0.0380369i	−0.854836	−	0.518898i	\(-0.826342\pi\)
							0.876797	+	0.480861i	\(0.159676\pi\)
\(47\)	5.31663	−	5.31663i	0.775511	−	0.775511i	−0.203553	−	0.979064i	\(-0.565249\pi\)
							0.979064	+	0.203553i	\(0.0652490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bf.a.67.17	✓	280
4.3	odd	2	inner	572.2.bf.a.67.28	yes	280
13.7	odd	12	inner	572.2.bf.a.111.28	yes	280
52.7	even	12	inner	572.2.bf.a.111.17	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bf.a.67.17	✓	280	1.1	even	1	trivial
572.2.bf.a.67.28	yes	280	4.3	odd	2	inner
572.2.bf.a.111.17	yes	280	52.7	even	12	inner
572.2.bf.a.111.28	yes	280	13.7	odd	12	inner