Embedded newform 576.2.bm.a.13.18

• Label: 576.2.bm.a.13.18
• 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			13.18
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18500 + 0.771868i) q^{2} +(-0.449175 - 1.67279i) q^{3} +(0.808438 - 1.82932i) q^{4} +(0.169627 - 0.499705i) q^{5} +(1.82345 + 1.63555i) q^{6} +(-1.57668 + 2.05477i) q^{7} +(0.454001 + 2.79175i) q^{8} +(-2.59648 + 1.50275i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18500	+	0.771868i	−0.837920	+	0.545793i
							\(3\)	−0.449175	−	1.67279i	−0.259331	−	0.965788i
\(5\)	0.169627	−	0.499705i	0.0758594	−	0.223475i								−0.902385	−	0.430931i	\(-0.858185\pi\)
							0.978244	+	0.207456i	\(0.0665185\pi\)
\(7\)	−1.57668	+	2.05477i	−0.595928	+	0.776628i	−0.989789	−	0.142542i	\(-0.954473\pi\)
							0.393861	+	0.919170i	\(0.371139\pi\)
\(11\)	4.37501	+	0.286753i	1.31911	+	0.0864593i	0.708636	−	0.705574i	\(-0.249311\pi\)
							0.610478	+	0.792033i	\(0.290977\pi\)
\(13\)	−3.00694	+	3.42876i	−0.833975	+	0.950966i	−0.999378	−	0.0352601i	\(-0.988774\pi\)
							0.165403	+	0.986226i	\(0.447107\pi\)
\(17\)	−3.42677	−	3.42677i	−0.831113	−	0.831113i	0.156556	−	0.987669i	\(-0.449961\pi\)
							−0.987669	+	0.156556i	\(0.949961\pi\)
\(19\)	−0.675985	+	3.39840i	−0.155082	+	0.779647i	0.822447	+	0.568842i	\(0.192609\pi\)
							−0.977528	+	0.210805i	\(0.932391\pi\)
\(23\)	−4.35533	+	3.34196i	−0.908149	+	0.696847i	−0.953265	−	0.302134i	\(-0.902301\pi\)
							0.0451162	+	0.998982i	\(0.485634\pi\)
\(29\)	1.18435	−	2.40163i	0.219929	−	0.445972i	−0.758746	−	0.651387i	\(-0.774188\pi\)
							0.978675	+	0.205415i	\(0.0658543\pi\)
\(31\)	3.23242	+	1.86624i	0.580560	+	0.335187i	0.761356	−	0.648334i	\(-0.224534\pi\)
							−0.180796	+	0.983521i	\(0.557867\pi\)
\(37\)	1.61701	+	8.12927i	0.265835	+	1.33644i	0.850845	+	0.525418i	\(0.176091\pi\)
							−0.585009	+	0.811027i	\(0.698909\pi\)
\(41\)	7.93421	−	6.08813i	1.23912	−	0.950806i	0.239318	−	0.970941i	\(-0.423076\pi\)
							0.999797	+	0.0201349i	\(0.00640957\pi\)
\(43\)	−0.343437	+	5.23984i	−0.0523737	+	0.799068i	0.888195	+	0.459466i	\(0.151959\pi\)
							−0.940569	+	0.339602i	\(0.889707\pi\)
\(47\)	−1.76127	+	6.57313i	−0.256907	+	0.958790i	0.710113	+	0.704088i	\(0.248644\pi\)
							−0.967020	+	0.254702i	\(0.918023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.18	✓	1504
9.7	even	3	inner	576.2.bm.a.205.80	yes	1504
64.5	even	16	inner	576.2.bm.a.517.80	yes	1504
576.133	even	48	inner	576.2.bm.a.133.18	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.18	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.18	yes	1504	576.133	even	48	inner
576.2.bm.a.205.80	yes	1504	9.7	even	3	inner
576.2.bm.a.517.80	yes	1504	64.5	even	16	inner