Embedded newform 576.2.bm.a.13.5

• Label: 576.2.bm.a.13.5
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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.bm (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			13.5
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40630 - 0.149384i) q^{2} +(-1.38997 + 1.03343i) q^{3} +(1.95537 + 0.420158i) q^{4} +(-0.631028 + 1.85895i) q^{5} +(2.10910 - 1.24568i) q^{6} +(0.0625219 - 0.0814801i) q^{7} +(-2.68707 - 0.882971i) q^{8} +(0.864037 - 2.87288i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40630	−	0.149384i	−0.994405	−	0.105631i
							\(3\)	−1.38997	+	1.03343i	−0.802500	+	0.596652i
\(5\)	−0.631028	+	1.85895i	−0.282204	+	0.831347i								0.709717	+	0.704487i	\(0.248823\pi\)
							−0.991921	+	0.126859i	\(0.959510\pi\)
\(7\)	0.0625219	−	0.0814801i	0.0236311	−	0.0307966i	−0.781388	−	0.624045i	\(-0.785488\pi\)
							0.805019	+	0.593248i	\(0.202155\pi\)
\(11\)	4.29865	+	0.281748i	1.29609	+	0.0849503i	0.697779	−	0.716313i	\(-0.254172\pi\)
							0.598312	+	0.801263i	\(0.295838\pi\)
\(13\)	2.27405	−	2.59306i	0.630709	−	0.719185i	−0.344916	−	0.938634i	\(-0.612092\pi\)
							0.975624	+	0.219448i	\(0.0704258\pi\)
\(17\)	4.05406	+	4.05406i	0.983254	+	0.983254i	0.999862	−	0.0166082i	\(-0.00528679\pi\)
							−0.0166082	+	0.999862i	\(0.505287\pi\)
\(19\)	1.11476	−	5.60428i	0.255744	−	1.28571i	−0.612856	−	0.790195i	\(-0.709979\pi\)
							0.868599	−	0.495515i	\(-0.165021\pi\)
\(23\)	−6.56859	+	5.04026i	−1.36965	+	1.05097i	−0.377483	+	0.926016i	\(0.623210\pi\)
							−0.992163	+	0.124950i	\(0.960123\pi\)
\(29\)	2.31638	−	4.69716i	0.430141	−	0.872240i	−0.568626	−	0.822596i	\(-0.692525\pi\)
							0.998767	−	0.0496437i	\(-0.0158086\pi\)
\(31\)	−1.41411	−	0.816435i	−0.253981	−	0.146636i	0.367605	−	0.929982i	\(-0.380178\pi\)
							−0.621586	+	0.783346i	\(0.713511\pi\)
\(37\)	1.60924	+	8.09020i	0.264558	+	1.33002i	0.853182	+	0.521614i	\(0.174670\pi\)
							−0.588624	+	0.808407i	\(0.700330\pi\)
\(41\)	−2.70826	+	2.07812i	−0.422960	+	0.324548i	−0.798215	−	0.602373i	\(-0.794222\pi\)
							0.375255	+	0.926922i	\(0.377555\pi\)
\(43\)	0.206287	−	3.14734i	0.0314585	−	0.479965i	−0.952842	−	0.303466i	\(-0.901856\pi\)
							0.984301	−	0.176499i	\(-0.0564772\pi\)
\(47\)	−1.82199	+	6.79974i	−0.265764	+	0.991845i	0.696017	+	0.718025i	\(0.254954\pi\)
							−0.961781	+	0.273819i	\(0.911713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.5	✓	1504
9.7	even	3	inner	576.2.bm.a.205.60	yes	1504
64.5	even	16	inner	576.2.bm.a.517.60	yes	1504
576.133	even	48	inner	576.2.bm.a.133.5	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.5	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.5	yes	1504	576.133	even	48	inner
576.2.bm.a.205.60	yes	1504	9.7	even	3	inner
576.2.bm.a.517.60	yes	1504	64.5	even	16	inner