Embedded newform 572.2.bs.a.15.13

• Label: 572.2.bs.a.15.13
• Level: $572$
• Weight: $2$
• Character: 572.15
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(80\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			15.13
Character	\(\chi\)	\(=\)	572.15
Dual form			572.2.bs.a.267.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25157 - 0.658464i) q^{2} +(0.151963 + 0.714928i) q^{3} +(1.13285 + 1.64823i) q^{4} +(0.124235 - 0.784389i) q^{5} +(0.280563 - 0.994844i) q^{6} +(2.56674 - 1.66686i) q^{7} +(-0.332541 - 2.80881i) q^{8} +(2.25261 - 1.00293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 12 q^{2} - 18 q^{4} - 24 q^{5} - 4 q^{6} - 12 q^{8} - 156 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25157	−	0.658464i	−0.884993	−	0.465605i
							\(3\)	0.151963	+	0.714928i	0.0877357	+	0.412764i	0.999995	+	0.00315081i	\(0.00100294\pi\)
−0.912259	+	0.409613i	\(0.865664\pi\)
\(5\)	0.124235	−	0.784389i	0.0555596	−	0.350790i	−0.944210	−	0.329343i	\(-0.893173\pi\)
							0.999770	−	0.0214467i	\(-0.00682722\pi\)
\(7\)	2.56674	−	1.66686i	0.970136	−	0.630013i	0.0407678	−	0.999169i	\(-0.487020\pi\)
							0.929368	+	0.369155i	\(0.120353\pi\)
\(11\)	1.98759	−	2.65509i	0.599281	−	0.800539i
							\(13\)	−0.146280	+	3.60258i	−0.0405707	+	0.999177i
\(17\)	−7.91625	+	0.832032i	−1.91997	+	0.201797i								−0.987293	−	0.158909i	\(-0.949202\pi\)
							−0.932680	+	0.360706i	\(0.882536\pi\)
\(19\)	0.120022	−	2.29016i	0.0275349	−	0.525398i	−0.949690	−	0.313192i	\(-0.898602\pi\)
							0.977225	−	0.212206i	\(-0.0680649\pi\)
\(23\)	−2.55529	−	4.42589i	−0.532815	−	0.922862i	−0.999266	−	0.0383153i	\(-0.987801\pi\)
							0.466451	−	0.884547i	\(-0.345532\pi\)
\(29\)	2.98522	−	3.31542i	0.554341	−	0.615658i	−0.399221	−	0.916855i	\(-0.630719\pi\)
							0.953562	+	0.301196i	\(0.0973860\pi\)
\(31\)	−0.903362	−	5.70360i	−0.162249	−	1.02440i	−0.925625	−	0.378443i	\(-0.876460\pi\)
							0.763376	−	0.645954i	\(-0.223540\pi\)
\(37\)	3.54162	−	0.185608i	0.582238	−	0.0305138i	0.241056	−	0.970511i	\(-0.422506\pi\)
							0.341182	+	0.939997i	\(0.389173\pi\)
\(41\)	−0.315335	+	0.485574i	−0.0492471	+	0.0758339i	−0.862437	−	0.506165i	\(-0.831063\pi\)
							0.813189	+	0.581999i	\(0.197729\pi\)
\(43\)	2.66359	−	4.61347i	0.406194	−	0.703548i	−0.588266	−	0.808668i	\(-0.700189\pi\)
							0.994460	+	0.105119i	\(0.0335225\pi\)
\(47\)	5.02190	+	9.85604i	0.732520	+	1.43765i	0.892741	+	0.450570i	\(0.148779\pi\)
							−0.160221	+	0.987081i	\(0.551221\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bs.a.15.13	✓	1280
4.3	odd	2	inner	572.2.bs.a.15.34	yes	1280
11.3	even	5	inner	572.2.bs.a.223.22	yes	1280
13.7	odd	12	inner	572.2.bs.a.59.2	yes	1280
44.3	odd	10	inner	572.2.bs.a.223.2	yes	1280
52.7	even	12	inner	572.2.bs.a.59.22	yes	1280
143.124	odd	60	inner	572.2.bs.a.267.34	yes	1280
572.267	even	60	inner	572.2.bs.a.267.13	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bs.a.15.13	✓	1280	1.1	even	1	trivial
572.2.bs.a.15.34	yes	1280	4.3	odd	2	inner
572.2.bs.a.59.2	yes	1280	13.7	odd	12	inner
572.2.bs.a.59.22	yes	1280	52.7	even	12	inner
572.2.bs.a.223.2	yes	1280	44.3	odd	10	inner
572.2.bs.a.223.22	yes	1280	11.3	even	5	inner
572.2.bs.a.267.13	yes	1280	572.267	even	60	inner
572.2.bs.a.267.34	yes	1280	143.124	odd	60	inner