Embedded newform 576.2.bm.a.13.94

• Label: 576.2.bm.a.13.94
• 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			13.94
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.94

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41194 - 0.0801391i) q^{2} +(-1.17907 - 1.26878i) q^{3} +(1.98716 - 0.226303i) q^{4} +(-0.483963 + 1.42571i) q^{5} +(-1.76645 - 1.69695i) q^{6} +(0.931501 - 1.21396i) q^{7} +(2.78761 - 0.478776i) q^{8} +(-0.219601 + 2.99195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41194	−	0.0801391i	0.998393	−	0.0566669i
							\(3\)	−1.17907	−	1.26878i	−0.680735	−	0.732530i
\(5\)	−0.483963	+	1.42571i	−0.216435	+	0.637596i								0.783435	+	0.621473i	\(0.213466\pi\)
							−0.999870	+	0.0161227i	\(0.994868\pi\)
\(7\)	0.931501	−	1.21396i	0.352074	−	0.458832i	−0.583422	−	0.812169i	\(-0.698287\pi\)
							0.935496	+	0.353337i	\(0.114953\pi\)
\(11\)	4.18146	+	0.274067i	1.26076	+	0.0826344i	0.681078	−	0.732210i	\(-0.261511\pi\)
							0.579679	+	0.814845i	\(0.303178\pi\)
\(13\)	1.51360	−	1.72593i	0.419798	−	0.478688i	−0.502827	−	0.864387i	\(-0.667707\pi\)
							0.922624	+	0.385700i	\(0.126040\pi\)
\(17\)	−4.20334	−	4.20334i	−1.01946	−	1.01946i	−0.999807	−	0.0196524i	\(-0.993744\pi\)
							−0.0196524	−	0.999807i	\(-0.506256\pi\)
\(19\)	0.0676413	−	0.340056i	0.0155180	−	0.0780141i	−0.972250	−	0.233943i	\(-0.924837\pi\)
							0.987768	+	0.155929i	\(0.0498371\pi\)
\(23\)	0.701369	−	0.538179i	0.146245	−	0.112218i	−0.533047	−	0.846085i	\(-0.678953\pi\)
							0.679293	+	0.733867i	\(0.262287\pi\)
\(29\)	−0.678962	+	1.37680i	−0.126080	+	0.255665i	−0.950756	−	0.309940i	\(-0.899691\pi\)
							0.824676	+	0.565606i	\(0.191358\pi\)
\(31\)	3.11702	+	1.79961i	0.559834	+	0.323220i	0.753079	−	0.657930i	\(-0.228568\pi\)
							−0.193245	+	0.981151i	\(0.561901\pi\)
\(37\)	−1.96962	−	9.90194i	−0.323803	−	1.62787i	−0.709122	−	0.705086i	\(-0.750908\pi\)
							0.385318	−	0.922784i	\(-0.374092\pi\)
\(41\)	−8.86630	+	6.80335i	−1.38468	+	1.06251i	−0.395086	+	0.918644i	\(0.629285\pi\)
							−0.989598	+	0.143861i	\(0.954048\pi\)
\(43\)	0.503150	−	7.67659i	0.0767297	−	1.17067i	−0.769785	−	0.638303i	\(-0.779637\pi\)
							0.846515	−	0.532365i	\(-0.178697\pi\)
\(47\)	−1.45195	+	5.41876i	−0.211789	+	0.790408i	0.775483	+	0.631368i	\(0.217506\pi\)
							−0.987272	+	0.159039i	\(0.949160\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.94	✓	1504
9.7	even	3	inner	576.2.bm.a.205.31	yes	1504
64.5	even	16	inner	576.2.bm.a.517.31	yes	1504
576.133	even	48	inner	576.2.bm.a.133.94	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.94	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.94	yes	1504	576.133	even	48	inner
576.2.bm.a.205.31	yes	1504	9.7	even	3	inner
576.2.bm.a.517.31	yes	1504	64.5	even	16	inner