Embedded newform 576.2.bm.a.517.83

• Label: 576.2.bm.a.517.83
• 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			517.83
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.83

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30543 + 0.543929i) q^{2} +(1.15779 - 1.28823i) q^{3} +(1.40828 + 1.42012i) q^{4} +(-0.183842 + 0.209632i) q^{5} +(2.21211 - 1.05194i) q^{6} +(1.21328 - 0.159731i) q^{7} +(1.06596 + 2.61987i) q^{8} +(-0.319065 - 2.98298i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30543	+	0.543929i	0.923077	+	0.384616i
							\(3\)	1.15779	−	1.28823i	0.668448	−	0.743759i
\(5\)	−0.183842	+	0.209632i	−0.0822168	+	0.0937503i								−0.791486	−	0.611188i	\(-0.790692\pi\)
							0.709269	+	0.704938i	\(0.249025\pi\)
\(7\)	1.21328	−	0.159731i	0.458575	−	0.0603725i	0.102300	−	0.994754i	\(-0.467380\pi\)
							0.356275	+	0.934381i	\(0.384047\pi\)
\(11\)	−0.510235	+	1.03465i	−0.153842	+	0.311960i	−0.960126	−	0.279568i	\(-0.909809\pi\)
							0.806284	+	0.591529i	\(0.201475\pi\)
\(13\)	−0.0481770	+	0.141925i	−0.0133619	+	0.0393629i	−0.953407	−	0.301689i	\(-0.902450\pi\)
							0.940045	+	0.341051i	\(0.110783\pi\)
\(17\)	2.67303	−	2.67303i	0.648306	−	0.648306i	−0.304277	−	0.952583i	\(-0.598415\pi\)
							0.952583	+	0.304277i	\(0.0984150\pi\)
\(19\)	0.709222	+	3.56550i	0.162707	+	0.817982i	0.972794	+	0.231672i	\(0.0744197\pi\)
							−0.810087	+	0.586309i	\(0.800580\pi\)
\(23\)	0.854444	−	6.49015i	0.178164	−	1.35329i	−0.635566	−	0.772046i	\(-0.719233\pi\)
							0.813730	−	0.581243i	\(-0.197433\pi\)
\(29\)	−5.01085	−	0.328428i	−0.930491	−	0.0609876i	−0.407386	−	0.913256i	\(-0.633560\pi\)
							−0.523105	+	0.852268i	\(0.675226\pi\)
\(31\)	−7.19242	−	4.15254i	−1.29180	−	0.745819i	−0.312823	−	0.949811i	\(-0.601275\pi\)
							−0.978972	+	0.203993i	\(0.934608\pi\)
\(37\)	0.345024	−	1.73455i	0.0567216	−	0.285159i	−0.942006	−	0.335597i	\(-0.891062\pi\)
							0.998727	+	0.0504384i	\(0.0160619\pi\)
\(41\)	−0.636262	+	4.83289i	−0.0993674	+	0.754770i	0.867128	+	0.498086i	\(0.165964\pi\)
							−0.966495	+	0.256685i	\(0.917370\pi\)
\(43\)	5.19198	+	2.56040i	0.791769	+	0.390457i	0.792754	−	0.609542i	\(-0.208647\pi\)
							−0.000984401		1.00000i	\(0.500313\pi\)
\(47\)	−7.62894	−	2.04417i	−1.11279	−	0.298173i	−0.344831	−	0.938665i	\(-0.612064\pi\)
							−0.767964	+	0.640492i	\(0.778730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.83	yes	1504
9.7	even	3	inner	576.2.bm.a.133.43	yes	1504
64.13	even	16	inner	576.2.bm.a.13.43	✓	1504
576.205	even	48	inner	576.2.bm.a.205.83	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.43	✓	1504	64.13	even	16	inner
576.2.bm.a.133.43	yes	1504	9.7	even	3	inner
576.2.bm.a.205.83	yes	1504	576.205	even	48	inner
576.2.bm.a.517.83	yes	1504	1.1	even	1	trivial