Embedded newform 576.2.bm.a.13.81

• Label: 576.2.bm.a.13.81
• 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			13.81
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.81

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25975 + 0.642671i) q^{2} +(0.944781 - 1.45168i) q^{3} +(1.17395 + 1.61921i) q^{4} +(-0.0403740 + 0.118938i) q^{5} +(2.12314 - 1.22158i) q^{6} +(2.88200 - 3.75590i) q^{7} +(0.438263 + 2.79427i) q^{8} +(-1.21478 - 2.74305i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25975	+	0.642671i	0.890779	+	0.454437i
							\(3\)	0.944781	−	1.45168i	0.545470	−	0.838131i
\(5\)	−0.0403740	+	0.118938i	−0.0180558	+	0.0531906i								−0.955584	−	0.294717i	\(-0.904775\pi\)
							0.937529	+	0.347908i	\(0.113108\pi\)
\(7\)	2.88200	−	3.75590i	1.08929	−	1.41960i	0.190251	−	0.981736i	\(-0.439070\pi\)
							0.899043	−	0.437860i	\(-0.144263\pi\)
\(11\)	1.43165	+	0.0938351i	0.431658	+	0.0282924i	0.279684	−	0.960092i	\(-0.409770\pi\)
							0.151974	+	0.988384i	\(0.451437\pi\)
\(13\)	−3.64121	+	4.15200i	−1.00989	+	1.15156i	−0.0214769	+	0.999769i	\(0.506837\pi\)
							−0.988413	+	0.151789i	\(0.951497\pi\)
\(17\)	−4.39730	−	4.39730i	−1.06650	−	1.06650i	−0.997625	−	0.0688771i	\(-0.978058\pi\)
							−0.0688771	−	0.997625i	\(-0.521942\pi\)
\(19\)	−0.571995	+	2.87561i	−0.131225	+	0.659710i	0.858041	+	0.513581i	\(0.171682\pi\)
							−0.989266	+	0.146129i	\(0.953318\pi\)
\(23\)	2.29803	−	1.76334i	0.479172	−	0.367682i	−0.340752	−	0.940153i	\(-0.610682\pi\)
							0.819925	+	0.572471i	\(0.194015\pi\)
\(29\)	2.36458	−	4.79489i	0.439091	−	0.890389i	−0.559031	−	0.829146i	\(-0.688827\pi\)
							0.998123	−	0.0612430i	\(-0.0195064\pi\)
\(31\)	0.790492	+	0.456391i	0.141977	+	0.0819702i	0.569306	−	0.822126i	\(-0.307212\pi\)
							−0.427329	+	0.904096i	\(0.640545\pi\)
\(37\)	1.83221	+	9.21117i	0.301214	+	1.51431i	0.774032	+	0.633146i	\(0.218237\pi\)
							−0.472818	+	0.881160i	\(0.656763\pi\)
\(41\)	−3.67149	+	2.81723i	−0.573390	+	0.439978i	−0.854318	−	0.519751i	\(-0.826025\pi\)
							0.280928	+	0.959729i	\(0.409358\pi\)
\(43\)	−0.461843	+	7.04636i	−0.0704304	+	1.07456i	0.805649	+	0.592393i	\(0.201817\pi\)
							−0.876079	+	0.482167i	\(0.839850\pi\)
\(47\)	−1.74337	+	6.50633i	−0.254296	+	0.949046i	0.714185	+	0.699957i	\(0.246798\pi\)
							−0.968481	+	0.249089i	\(0.919869\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.81	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.81	yes	1504	576.133	even	48	inner
576.2.bm.a.205.45	yes	1504	9.7	even	3	inner
576.2.bm.a.517.45	yes	1504	64.5	even	16	inner