Embedded newform 576.2.bm.a.205.71

• Label: 576.2.bm.a.205.71
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.71
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.71

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03047 - 0.968568i) q^{2} +(1.63705 - 0.565742i) q^{3} +(0.123751 - 1.99617i) q^{4} +(-0.289353 - 0.329944i) q^{5} +(1.13898 - 2.16858i) q^{6} +(-0.132159 - 0.0173990i) q^{7} +(-1.80590 - 2.17686i) q^{8} +(2.35987 - 1.85230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03047	−	0.968568i	0.728655	−	0.684881i
							\(3\)	1.63705	−	0.565742i	0.945152	−	0.326631i
\(5\)	−0.289353	−	0.329944i	−0.129403	−	0.147555i								0.683510	−	0.729941i	\(-0.260453\pi\)
							−0.812912	+	0.582386i	\(0.802119\pi\)
\(7\)	−0.132159	−	0.0173990i	−0.0499512	−	0.00657621i	0.105509	−	0.994418i	\(-0.466353\pi\)
							−0.155460	+	0.987842i	\(0.549686\pi\)
\(11\)	−2.45699	−	4.98228i	−0.740811	−	1.50222i	−0.860526	−	0.509407i	\(-0.829865\pi\)
							0.119715	−	0.992808i	\(-0.461802\pi\)
\(13\)	1.43540	+	4.22854i	0.398107	+	1.17279i	0.942572	+	0.334002i	\(0.108399\pi\)
							−0.544465	+	0.838783i	\(0.683267\pi\)
\(17\)	0.861641	+	0.861641i	0.208979	+	0.208979i	0.803833	−	0.594855i	\(-0.202790\pi\)
							−0.594855	+	0.803833i	\(0.702790\pi\)
\(19\)	−1.11562	+	5.60861i	−0.255941	+	1.28670i	0.612326	+	0.790605i	\(0.290234\pi\)
							−0.868267	+	0.496097i	\(0.834766\pi\)
\(23\)	0.478102	+	3.63155i	0.0996912	+	0.757230i	0.966150	+	0.257981i	\(0.0830572\pi\)
							−0.866459	+	0.499249i	\(0.833609\pi\)
\(29\)	5.16048	−	0.338235i	0.958276	−	0.0628087i	0.421768	−	0.906704i	\(-0.361410\pi\)
							0.536508	+	0.843895i	\(0.319743\pi\)
\(31\)	6.68600	−	3.86017i	1.20084	−	0.693306i	0.240099	−	0.970748i	\(-0.422820\pi\)
							0.960742	+	0.277442i	\(0.0894867\pi\)
\(37\)	0.738679	+	3.71359i	0.121438	+	0.610510i	0.992791	+	0.119855i	\(0.0382428\pi\)
							−0.871353	+	0.490656i	\(0.836757\pi\)
\(41\)	0.487315	+	3.70153i	0.0761059	+	0.578082i	0.986430	+	0.164185i	\(0.0524993\pi\)
							−0.910324	+	0.413897i	\(0.864167\pi\)
\(43\)	−3.01870	+	1.48866i	−0.460347	+	0.227018i	−0.657649	−	0.753325i	\(-0.728449\pi\)
							0.197302	+	0.980343i	\(0.436782\pi\)
\(47\)	−1.60306	+	0.429538i	−0.233830	+	0.0626545i	−0.373831	−	0.927497i	\(-0.621956\pi\)
							0.140001	+	0.990151i	\(0.455289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.71	yes	1504
9.4	even	3	inner	576.2.bm.a.13.55	✓	1504
64.5	even	16	inner	576.2.bm.a.133.55	yes	1504
576.517	even	48	inner	576.2.bm.a.517.71	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.55	✓	1504	9.4	even	3	inner
576.2.bm.a.133.55	yes	1504	64.5	even	16	inner
576.2.bm.a.205.71	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.71	yes	1504	576.517	even	48	inner