Embedded newform 576.2.bm.a.205.5

• Label: 576.2.bm.a.205.5
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.5
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39881 + 0.208169i) q^{2} +(0.0768748 - 1.73034i) q^{3} +(1.91333 - 0.582377i) q^{4} +(-2.08768 - 2.38054i) q^{5} +(0.252671 + 2.43642i) q^{6} +(2.53604 + 0.333876i) q^{7} +(-2.55515 + 1.21293i) q^{8} +(-2.98818 - 0.266040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39881	+	0.208169i	−0.989107	+	0.147198i
							\(3\)	0.0768748	−	1.73034i	0.0443837	−	0.999015i
\(5\)	−2.08768	−	2.38054i	−0.933639	−	1.06461i								−0.997775	−	0.0666705i	\(-0.978762\pi\)
							0.0641356	−	0.997941i	\(-0.479571\pi\)
\(7\)	2.53604	+	0.333876i	0.958533	+	0.126193i	0.593529	−	0.804813i	\(-0.297734\pi\)
							0.365004	+	0.931006i	\(0.381068\pi\)
\(11\)	−0.249733	−	0.506409i	−0.0752973	−	0.152688i	0.855969	−	0.517027i	\(-0.172961\pi\)
							−0.931266	+	0.364339i	\(0.881295\pi\)
\(13\)	−1.58540	−	4.67044i	−0.439711	−	1.29535i	−0.910649	−	0.413181i	\(-0.864418\pi\)
							0.470938	−	0.882167i	\(-0.343916\pi\)
\(17\)	1.15462	+	1.15462i	0.280037	+	0.280037i	0.833124	−	0.553087i	\(-0.186550\pi\)
							−0.553087	+	0.833124i	\(0.686550\pi\)
\(19\)	0.747136	−	3.75611i	0.171405	−	0.861710i	−0.795379	−	0.606113i	\(-0.792728\pi\)
							0.966783	−	0.255597i	\(-0.0822720\pi\)
\(23\)	0.00142803	+	0.0108469i	0.000297764	+	0.00226174i	0.991593	−	0.129395i	\(-0.0413036\pi\)
							−0.991295	+	0.131657i	\(0.957970\pi\)
\(29\)	−4.49301	+	0.294488i	−0.834332	+	0.0546850i	−0.476579	−	0.879132i	\(-0.658123\pi\)
							−0.357752	+	0.933816i	\(0.616457\pi\)
\(31\)	−3.52770	+	2.03672i	−0.633594	+	0.365805i	−0.782142	−	0.623100i	\(-0.785873\pi\)
							0.148549	+	0.988905i	\(0.452540\pi\)
\(37\)	0.974818	+	4.90074i	0.160259	+	0.805677i	0.974367	+	0.224963i	\(0.0722262\pi\)
							−0.814108	+	0.580713i	\(0.802774\pi\)
\(41\)	0.378647	+	2.87611i	0.0591347	+	0.449173i	0.995240	+	0.0974532i	\(0.0310696\pi\)
							−0.936105	+	0.351720i	\(0.885597\pi\)
\(43\)	−6.76560	+	3.33643i	−1.03174	+	0.508800i	−0.877673	−	0.479259i	\(-0.840905\pi\)
							−0.154072	+	0.988060i	\(0.549239\pi\)
\(47\)	3.55593	−	0.952808i	0.518685	−	0.138981i	0.0100269	−	0.999950i	\(-0.496808\pi\)
							0.508658	+	0.860968i	\(0.330142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.5	yes	1504
9.4	even	3	inner	576.2.bm.a.13.59	✓	1504
64.5	even	16	inner	576.2.bm.a.133.59	yes	1504
576.517	even	48	inner	576.2.bm.a.517.5	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.59	✓	1504	9.4	even	3	inner
576.2.bm.a.133.59	yes	1504	64.5	even	16	inner
576.2.bm.a.205.5	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.5	yes	1504	576.517	even	48	inner