Embedded newform 576.2.bm.a.517.12

• Label: 576.2.bm.a.517.12
• 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			517.12
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29352 - 0.571665i) q^{2} +(1.48784 + 0.886751i) q^{3} +(1.34640 + 1.47892i) q^{4} +(-2.64975 + 3.02146i) q^{5} +(-1.41763 - 1.99758i) q^{6} +(-4.44635 + 0.585373i) q^{7} +(-0.896148 - 2.68271i) q^{8} +(1.42734 + 2.63869i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29352	−	0.571665i	−0.914658	−	0.404228i
							\(3\)	1.48784	+	0.886751i	0.859006	+	0.511966i
\(5\)	−2.64975	+	3.02146i	−1.18501	+	1.35124i								−0.262652	+	0.964891i	\(0.584597\pi\)
							−0.922353	+	0.386349i	\(0.873736\pi\)
\(7\)	−4.44635	+	0.585373i	−1.68056	+	0.221250i	−0.909557	−	0.415579i	\(-0.863579\pi\)
							−0.771005	+	0.636829i	\(0.780246\pi\)
\(11\)	−0.255650	+	0.518408i	−0.0770815	+	0.156306i	−0.932002	−	0.362453i	\(-0.881939\pi\)
							0.854920	+	0.518759i	\(0.173606\pi\)
\(13\)	2.16473	−	6.37708i	0.600387	−	1.76868i	−0.0383580	−	0.999264i	\(-0.512213\pi\)
							0.638745	−	0.769419i	\(-0.279454\pi\)
\(17\)	−1.08758	+	1.08758i	−0.263776	+	0.263776i	−0.826586	−	0.562810i	\(-0.809720\pi\)
							0.562810	+	0.826586i	\(0.309720\pi\)
\(19\)	−0.700018	−	3.51923i	−0.160595	−	0.807366i	−0.974154	−	0.225884i	\(-0.927473\pi\)
							0.813559	−	0.581482i	\(-0.197527\pi\)
\(23\)	−0.182415	+	1.38558i	−0.0380361	+	0.288913i	0.961846	+	0.273590i	\(0.0882111\pi\)
							−0.999883	+	0.0153233i	\(0.995122\pi\)
\(29\)	−4.44111	−	0.291086i	−0.824693	−	0.0540532i	−0.352789	−	0.935703i	\(-0.614767\pi\)
							−0.471904	+	0.881650i	\(0.656433\pi\)
\(31\)	−1.28083	−	0.739487i	−0.230044	−	0.132816i	0.380549	−	0.924761i	\(-0.375735\pi\)
							−0.610592	+	0.791945i	\(0.709069\pi\)
\(37\)	−0.465852	+	2.34200i	−0.0765856	+	0.385022i	0.923414	+	0.383806i	\(0.125387\pi\)
							−0.999999	+	0.00121571i	\(0.999613\pi\)
\(41\)	−0.0670228	+	0.509089i	−0.0104672	+	0.0795064i	−0.995881	−	0.0906680i	\(-0.971100\pi\)
							0.985414	+	0.170174i	\(0.0544331\pi\)
\(43\)	2.60490	+	1.28460i	0.397244	+	0.195899i	0.629928	−	0.776653i	\(-0.283084\pi\)
							−0.232684	+	0.972552i	\(0.574751\pi\)
\(47\)	−5.59094	−	1.49809i	−0.815522	−	0.218518i	−0.173134	−	0.984898i	\(-0.555389\pi\)
							−0.642388	+	0.766380i	\(0.722056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.12	yes	1504
9.7	even	3	inner	576.2.bm.a.133.50	yes	1504
64.13	even	16	inner	576.2.bm.a.13.50	✓	1504
576.205	even	48	inner	576.2.bm.a.205.12	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.50	✓	1504	64.13	even	16	inner
576.2.bm.a.133.50	yes	1504	9.7	even	3	inner
576.2.bm.a.205.12	yes	1504	576.205	even	48	inner
576.2.bm.a.517.12	yes	1504	1.1	even	1	trivial