Embedded newform 576.2.bm.a.517.65

• Label: 576.2.bm.a.517.65
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.65
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.65

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.782397 + 1.17807i) q^{2} +(-0.422226 - 1.67980i) q^{3} +(-0.775711 + 1.84344i) q^{4} +(-0.0860861 + 0.0981624i) q^{5} +(1.64858 - 1.81168i) q^{6} +(-3.69234 + 0.486106i) q^{7} +(-2.77862 + 0.528457i) q^{8} +(-2.64345 + 1.41851i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.782397	+	1.17807i	0.553238	+	0.833023i
							\(3\)	−0.422226	−	1.67980i	−0.243772	−	0.969832i
\(5\)	−0.0860861	+	0.0981624i	−0.0384989	+	0.0438996i								−0.770769	−	0.637115i	\(-0.780128\pi\)
							0.732270	+	0.681015i	\(0.238461\pi\)
\(7\)	−3.69234	+	0.486106i	−1.39557	+	0.183731i	−0.790511	−	0.612448i	\(-0.790185\pi\)
							−0.605062	+	0.796178i	\(0.706852\pi\)
\(11\)	2.51386	−	5.09761i	0.757957	−	1.53699i	−0.0834057	−	0.996516i	\(-0.526580\pi\)
							0.841363	−	0.540470i	\(-0.181754\pi\)
\(13\)	2.15615	−	6.35182i	0.598009	−	1.76168i	−0.0485573	−	0.998820i	\(-0.515462\pi\)
							0.646566	−	0.762858i	\(-0.276204\pi\)
\(17\)	−0.490212	+	0.490212i	−0.118894	+	0.118894i	−0.764050	−	0.645157i	\(-0.776792\pi\)
							0.645157	+	0.764050i	\(0.276792\pi\)
\(19\)	−0.255708	−	1.28553i	−0.0586635	−	0.294921i	0.940305	−	0.340333i	\(-0.110540\pi\)
							−0.998968	+	0.0454120i	\(0.985540\pi\)
\(23\)	0.753805	−	5.72572i	0.157179	−	1.19389i	−0.713002	−	0.701162i	\(-0.752665\pi\)
							0.870181	−	0.492732i	\(-0.164002\pi\)
\(29\)	−7.24329	−	0.474750i	−1.34504	−	0.0881589i	−0.624205	−	0.781261i	\(-0.714577\pi\)
							−0.720840	+	0.693102i	\(0.756244\pi\)
\(31\)	−4.69271	−	2.70934i	−0.842836	−	0.486611i	0.0153915	−	0.999882i	\(-0.495101\pi\)
							−0.858227	+	0.513270i	\(0.828434\pi\)
\(37\)	0.531839	−	2.67373i	0.0874338	−	0.439559i	−0.912127	−	0.409909i	\(-0.865561\pi\)
							0.999560	−	0.0296505i	\(-0.00943945\pi\)
\(41\)	−1.02879	+	7.81440i	−0.160669	+	1.22040i	0.701085	+	0.713078i	\(0.252699\pi\)
							−0.861754	+	0.507326i	\(0.830634\pi\)
\(43\)	−2.75289	−	1.35757i	−0.419812	−	0.207028i	0.220099	−	0.975478i	\(-0.429362\pi\)
							−0.639911	+	0.768449i	\(0.721029\pi\)
\(47\)	−3.95872	−	1.06074i	−0.577439	−	0.154724i	−0.0417330	−	0.999129i	\(-0.513288\pi\)
							−0.535706	+	0.844404i	\(0.679955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.65	yes	1504
9.7	even	3	inner	576.2.bm.a.133.62	yes	1504
64.13	even	16	inner	576.2.bm.a.13.62	✓	1504
576.205	even	48	inner	576.2.bm.a.205.65	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.62	✓	1504	64.13	even	16	inner
576.2.bm.a.133.62	yes	1504	9.7	even	3	inner
576.2.bm.a.205.65	yes	1504	576.205	even	48	inner
576.2.bm.a.517.65	yes	1504	1.1	even	1	trivial