Embedded newform 576.2.bl.a.419.15

• Label: 576.2.bl.a.419.15
• Level: $576$
• Weight: $2$
• Character: 576.419
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $1504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.bl (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(1504\)
Relative dimension:	\(94\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			419.15
Character	\(\chi\)	\(=\)	576.419
Dual form			576.2.bl.a.11.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24238 + 0.675648i) q^{2} +(1.24724 + 1.20182i) q^{3} +(1.08700 - 1.67882i) q^{4} +(-0.558818 + 0.275579i) q^{5} +(-2.36156 - 0.650421i) q^{6} +(-2.33627 + 3.04469i) q^{7} +(-0.216171 + 2.82015i) q^{8} +(0.111234 + 2.99794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24238	+	0.675648i	−0.878493	+	0.477755i
							\(3\)	1.24724	+	1.20182i	0.720097	+	0.693874i
\(5\)	−0.558818	+	0.275579i	−0.249911	+	0.123242i								−0.562931	−	0.826504i	\(-0.690326\pi\)
							0.313020	+	0.949747i	\(0.398659\pi\)
\(7\)	−2.33627	+	3.04469i	−0.883027	+	1.15078i	0.104686	+	0.994505i	\(0.466616\pi\)
							−0.987713	+	0.156278i	\(0.950050\pi\)
\(11\)	0.0326551	−	0.0286377i	0.00984587	−	0.00863460i	−0.654409	−	0.756141i	\(-0.727082\pi\)
							0.664255	+	0.747506i	\(0.268749\pi\)
\(13\)	6.07831	−	0.398394i	1.68582	−	0.110495i	0.807906	−	0.589311i	\(-0.200601\pi\)
							0.877915	+	0.478817i	\(0.158934\pi\)
\(17\)	−4.66084	+	4.66084i	−1.13042	+	1.13042i	−0.140312	+	0.990107i	\(0.544810\pi\)
							−0.990107	+	0.140312i	\(0.955190\pi\)
\(19\)	−3.07666	−	4.60455i	−0.705834	−	1.05636i	−0.995077	−	0.0991000i	\(-0.968404\pi\)
							0.289243	−	0.957256i	\(-0.406596\pi\)
\(23\)	−1.73814	−	2.26518i	−0.362427	−	0.472324i	0.576175	−	0.817326i	\(-0.304545\pi\)
							−0.938602	+	0.345003i	\(0.887878\pi\)
\(29\)	−0.252941	−	0.745140i	−0.0469699	−	0.138369i	0.920914	−	0.389765i	\(-0.127444\pi\)
							−0.967884	+	0.251396i	\(0.919110\pi\)
\(31\)	2.56818	−	4.44821i	0.461258	−	0.798922i	−0.537766	−	0.843094i	\(-0.680732\pi\)
							0.999024	+	0.0441718i	\(0.0140649\pi\)
\(37\)	6.90598	+	4.61443i	1.13534	+	0.758607i	0.973608	−	0.228228i	\(-0.0732932\pi\)
							0.161728	+	0.986835i	\(0.448293\pi\)
\(41\)	−8.87082	+	6.80682i	−1.38539	+	1.06305i	−0.395922	+	0.918284i	\(0.629575\pi\)
							−0.989466	+	0.144762i	\(0.953758\pi\)
\(43\)	−2.62907	+	2.30563i	−0.400929	+	0.351606i	−0.835941	−	0.548819i	\(-0.815078\pi\)
							0.435012	+	0.900425i	\(0.356744\pi\)
\(47\)	−1.26944	+	4.73762i	−0.185167	+	0.691053i	0.809428	+	0.587219i	\(0.199777\pi\)
							−0.994595	+	0.103833i	\(0.966889\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.419.15	yes	1504
9.2	odd	6	inner	576.2.bl.a.227.19	yes	1504
64.11	odd	16	inner	576.2.bl.a.203.19	yes	1504
576.11	even	48	inner	576.2.bl.a.11.15	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.15	✓	1504	576.11	even	48	inner
576.2.bl.a.203.19	yes	1504	64.11	odd	16	inner
576.2.bl.a.227.19	yes	1504	9.2	odd	6	inner
576.2.bl.a.419.15	yes	1504	1.1	even	1	trivial