Embedded newform 576.2.bm.a.517.88

• Label: 576.2.bm.a.517.88
• 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			517.88
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.88

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38225 - 0.298980i) q^{2} +(0.914527 + 1.47093i) q^{3} +(1.82122 - 0.826530i) q^{4} +(1.56322 - 1.78251i) q^{5} +(1.70388 + 1.75977i) q^{6} +(-0.280581 + 0.0369393i) q^{7} +(2.27026 - 1.68698i) q^{8} +(-1.32728 + 2.69041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38225	−	0.298980i	0.977397	−	0.211411i
							\(3\)	0.914527	+	1.47093i	0.528003	+	0.849243i
\(5\)	1.56322	−	1.78251i	0.699093	−	0.797163i								−0.287945	−	0.957647i	\(-0.592972\pi\)
							0.987038	+	0.160484i	\(0.0513054\pi\)
\(7\)	−0.280581	+	0.0369393i	−0.106050	+	0.0139617i	−0.183364	−	0.983045i	\(-0.558699\pi\)
							0.0773145	+	0.997007i	\(0.475365\pi\)
\(11\)	−0.394839	+	0.800655i	−0.119049	+	0.241407i	−0.948238	−	0.317562i	\(-0.897136\pi\)
							0.829189	+	0.558968i	\(0.188803\pi\)
\(13\)	−0.474075	+	1.39658i	−0.131485	+	0.387342i	−0.992656	−	0.120974i	\(-0.961398\pi\)
							0.861171	+	0.508315i	\(0.169732\pi\)
\(17\)	−3.08704	+	3.08704i	−0.748717	+	0.748717i	−0.974238	−	0.225522i	\(-0.927591\pi\)
							0.225522	+	0.974238i	\(0.427591\pi\)
\(19\)	−0.658744	−	3.31173i	−0.151126	−	0.759763i	−0.979792	−	0.200019i	\(-0.935900\pi\)
							0.828666	−	0.559744i	\(-0.189100\pi\)
\(23\)	1.11294	−	8.45365i	0.232065	−	1.76271i	−0.338035	−	0.941134i	\(-0.609762\pi\)
							0.570100	−	0.821575i	\(-0.306904\pi\)
\(29\)	−5.19229	−	0.340321i	−0.964184	−	0.0631960i	−0.424828	−	0.905274i	\(-0.639666\pi\)
							−0.539356	+	0.842078i	\(0.681332\pi\)
\(31\)	2.39713	+	1.38398i	0.430536	+	0.248570i	0.699575	−	0.714559i	\(-0.253373\pi\)
							−0.269039	+	0.963129i	\(0.586706\pi\)
\(37\)	−1.25131	+	6.29074i	−0.205713	+	1.03419i	0.730541	+	0.682869i	\(0.239268\pi\)
							−0.936254	+	0.351323i	\(0.885732\pi\)
\(41\)	−0.793302	+	6.02573i	−0.123893	+	0.941060i	0.810730	+	0.585420i	\(0.199070\pi\)
							−0.934623	+	0.355640i	\(0.884263\pi\)
\(43\)	1.25123	+	0.617040i	0.190811	+	0.0940978i	0.535185	−	0.844735i	\(-0.320242\pi\)
							−0.344373	+	0.938833i	\(0.611909\pi\)
\(47\)	4.62193	+	1.23844i	0.674178	+	0.180646i	0.579636	−	0.814875i	\(-0.303195\pi\)
							0.0945421	+	0.995521i	\(0.469861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.88	yes	1504
9.7	even	3	inner	576.2.bm.a.133.25	yes	1504
64.13	even	16	inner	576.2.bm.a.13.25	✓	1504
576.205	even	48	inner	576.2.bm.a.205.88	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.25	✓	1504	64.13	even	16	inner
576.2.bm.a.133.25	yes	1504	9.7	even	3	inner
576.2.bm.a.205.88	yes	1504	576.205	even	48	inner
576.2.bm.a.517.88	yes	1504	1.1	even	1	trivial