Embedded newform 572.2.bf.a.67.8

• Label: 572.2.bf.a.67.8
• 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			67.8
Character	\(\chi\)	\(=\)	572.67
Dual form			572.2.bf.a.111.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33591 + 0.464039i) q^{2} +(2.67338 - 1.54348i) q^{3} +(1.56934 - 1.23983i) q^{4} +(-1.23437 + 1.23437i) q^{5} +(-2.85518 + 3.30251i) q^{6} +(0.844074 + 3.15013i) q^{7} +(-1.52117 + 2.38454i) q^{8} +(3.26465 - 5.65455i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33591	+	0.464039i	−0.944634	+	0.328125i
							\(3\)	2.67338	−	1.54348i	1.54348	−	0.891128i	0.544863	−	0.838525i	\(-0.316581\pi\)
0.998615	−	0.0526032i	\(-0.0167519\pi\)
\(5\)	−1.23437	+	1.23437i	−0.552028	+	0.552028i	−0.927026	−	0.374998i	\(-0.877643\pi\)
							0.374998	+	0.927026i	\(0.377643\pi\)
\(7\)	0.844074	+	3.15013i	0.319030	+	1.19064i	0.920178	+	0.391499i	\(0.128043\pi\)
							−0.601148	+	0.799137i	\(0.705290\pi\)
\(11\)	0.965926	+	0.258819i	0.291238	+	0.0780369i
							\(13\)	3.49347	−	0.891999i	0.968914	−	0.247396i
\(17\)	−1.56935	−	0.906065i	−0.380623	−	0.219753i								0.297466	−	0.954732i	\(-0.403858\pi\)
							−0.678089	+	0.734979i	\(0.737192\pi\)
\(19\)	1.46339	−	0.392113i	0.335724	−	0.0899569i	−0.0870190	−	0.996207i	\(-0.527734\pi\)
							0.422743	+	0.906250i	\(0.361067\pi\)
\(23\)	0.623816	+	1.08048i	0.130075	+	0.225296i	0.923705	−	0.383104i	\(-0.125145\pi\)
							−0.793630	+	0.608400i	\(0.791812\pi\)
\(29\)	5.22126	+	9.04349i	0.969564	+	1.67933i	0.696818	+	0.717248i	\(0.254598\pi\)
							0.272746	+	0.962086i	\(0.412068\pi\)
\(31\)	−1.71419	−	1.71419i	−0.307878	−	0.307878i	0.536208	−	0.844086i	\(-0.319856\pi\)
							−0.844086	+	0.536208i	\(0.819856\pi\)
\(37\)	1.73453	−	6.47336i	0.285155	−	1.06421i	−0.663570	−	0.748114i	\(-0.730960\pi\)
							0.948726	−	0.316100i	\(-0.102374\pi\)
\(41\)	−8.85793	−	2.37347i	−1.38338	−	0.370674i	−0.511030	−	0.859563i	\(-0.670736\pi\)
							−0.872346	+	0.488889i	\(0.837402\pi\)
\(43\)	2.19837	−	3.80768i	0.335248	−	0.580666i	−0.648285	−	0.761398i	\(-0.724513\pi\)
							0.983532	+	0.180732i	\(0.0578467\pi\)
\(47\)	3.96207	−	3.96207i	0.577928	−	0.577928i	−0.356404	−	0.934332i	\(-0.615997\pi\)
							0.934332	+	0.356404i	\(0.115997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bf.a.67.8	✓	280
4.3	odd	2	inner	572.2.bf.a.67.20	yes	280
13.7	odd	12	inner	572.2.bf.a.111.20	yes	280
52.7	even	12	inner	572.2.bf.a.111.8	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bf.a.67.8	✓	280	1.1	even	1	trivial
572.2.bf.a.67.20	yes	280	4.3	odd	2	inner
572.2.bf.a.111.8	yes	280	52.7	even	12	inner
572.2.bf.a.111.20	yes	280	13.7	odd	12	inner