Embedded newform 576.2.bm.a.517.45

• Label: 576.2.bm.a.517.45
• 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			517.45
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0733064 + 1.41231i) q^{2} +(1.70273 + 0.317328i) q^{3} +(-1.98925 - 0.207063i) q^{4} +(-0.0828163 + 0.0944338i) q^{5} +(-0.572988 + 2.38153i) q^{6} +(-4.69370 + 0.617938i) q^{7} +(0.438263 - 2.79427i) q^{8} +(2.79861 + 1.08065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0733064	+	1.41231i	−0.0518355	+	0.998656i
							\(3\)	1.70273	+	0.317328i	0.983074	+	0.183209i
\(5\)	−0.0828163	+	0.0944338i	−0.0370366	+	0.0422321i								−0.770061	−	0.637970i	\(-0.779775\pi\)
							0.733025	+	0.680202i	\(0.238108\pi\)
\(7\)	−4.69370	+	0.617938i	−1.77405	+	0.233558i	−0.945333	−	0.326106i	\(-0.894263\pi\)
							−0.828719	+	0.559664i	\(0.810930\pi\)
\(11\)	−0.634560	+	1.28676i	−0.191327	+	0.387973i	−0.971306	−	0.237833i	\(-0.923563\pi\)
							0.779979	+	0.625806i	\(0.215230\pi\)
\(13\)	−1.77514	+	5.22938i	−0.492334	+	1.45037i	0.362753	+	0.931885i	\(0.381837\pi\)
							−0.855087	+	0.518484i	\(0.826497\pi\)
\(17\)	−4.39730	+	4.39730i	−1.06650	+	1.06650i	−0.0688771	+	0.997625i	\(0.521942\pi\)
							−0.997625	+	0.0688771i	\(0.978058\pi\)
\(19\)	−0.571995	−	2.87561i	−0.131225	−	0.659710i	−0.989266	−	0.146129i	\(-0.953318\pi\)
							0.858041	−	0.513581i	\(-0.171682\pi\)
\(23\)	0.378083	−	2.87182i	0.0788357	−	0.598816i	−0.905737	−	0.423840i	\(-0.860682\pi\)
							0.984573	−	0.174976i	\(-0.0559849\pi\)
\(29\)	−5.33479	−	0.349661i	−0.990646	−	0.0649303i	−0.438546	−	0.898709i	\(-0.644506\pi\)
							−0.552099	+	0.833778i	\(0.686173\pi\)
\(31\)	−0.790492	−	0.456391i	−0.141977	−	0.0819702i	0.427329	−	0.904096i	\(-0.359455\pi\)
							−0.569306	+	0.822126i	\(0.692788\pi\)
\(37\)	1.83221	−	9.21117i	0.301214	−	1.51431i	−0.472818	−	0.881160i	\(-0.656763\pi\)
							0.774032	−	0.633146i	\(-0.218237\pi\)
\(41\)	−0.604050	+	4.58822i	−0.0943368	+	0.716559i	0.877277	+	0.479985i	\(0.159358\pi\)
							−0.971613	+	0.236574i	\(0.923975\pi\)
\(43\)	6.33325	+	3.12321i	0.965812	+	0.476286i	0.855609	−	0.517623i	\(-0.173183\pi\)
							0.110203	+	0.993909i	\(0.464850\pi\)
\(47\)	6.50633	+	1.74337i	0.949046	+	0.254296i	0.699957	−	0.714185i	\(-0.253202\pi\)
							0.249089	+	0.968481i	\(0.419869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.45	yes	1504
9.7	even	3	inner	576.2.bm.a.133.81	yes	1504
64.13	even	16	inner	576.2.bm.a.13.81	✓	1504
576.205	even	48	inner	576.2.bm.a.205.45	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.81	✓	1504	64.13	even	16	inner
576.2.bm.a.133.81	yes	1504	9.7	even	3	inner
576.2.bm.a.205.45	yes	1504	576.205	even	48	inner
576.2.bm.a.517.45	yes	1504	1.1	even	1	trivial