Embedded newform 576.2.bm.a.205.89

• Label: 576.2.bm.a.205.89
• 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			205.89
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.89

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38975 - 0.261905i) q^{2} +(1.70060 + 0.328562i) q^{3} +(1.86281 - 0.727965i) q^{4} +(0.232101 + 0.264660i) q^{5} +(2.44946 + 0.0112231i) q^{6} +(-2.67566 - 0.352258i) q^{7} +(2.39819 - 1.49957i) q^{8} +(2.78409 + 1.11751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38975	−	0.261905i	0.982702	−	0.185195i
							\(3\)	1.70060	+	0.328562i	0.981843	+	0.189695i
\(5\)	0.232101	+	0.264660i	0.103799	+	0.118360i								0.801406	−	0.598120i	\(-0.204085\pi\)
							−0.697608	+	0.716480i	\(0.745752\pi\)
\(7\)	−2.67566	−	0.352258i	−1.01131	−	0.133141i	−0.393379	−	0.919376i	\(-0.628694\pi\)
							−0.617927	+	0.786235i	\(0.712027\pi\)
\(11\)	1.81344	+	3.67730i	0.546773	+	1.10875i	0.978405	+	0.206697i	\(0.0662713\pi\)
							−0.431632	+	0.902050i	\(0.642062\pi\)
\(13\)	0.566790	+	1.66971i	0.157199	+	0.463094i	0.996784	−	0.0801345i	\(-0.0255350\pi\)
							−0.839585	+	0.543229i	\(0.817202\pi\)
\(17\)	−4.32588	−	4.32588i	−1.04918	−	1.04918i	−0.998726	−	0.0504548i	\(-0.983933\pi\)
							−0.0504548	−	0.998726i	\(-0.516067\pi\)
\(19\)	−0.761827	+	3.82996i	−0.174775	+	0.878654i	0.789500	+	0.613751i	\(0.210340\pi\)
							−0.964275	+	0.264903i	\(0.914660\pi\)
\(23\)	−0.977017	−	7.42118i	−0.203722	−	1.54742i	−0.721808	−	0.692094i	\(-0.756688\pi\)
							0.518086	−	0.855329i	\(-0.326645\pi\)
\(29\)	−3.98656	+	0.261293i	−0.740286	+	0.0485209i	−0.430880	−	0.902409i	\(-0.641797\pi\)
							−0.309406	+	0.950930i	\(0.600130\pi\)
\(31\)	2.59062	−	1.49570i	0.465289	−	0.268635i	−0.248976	−	0.968510i	\(-0.580094\pi\)
							0.714266	+	0.699875i	\(0.246761\pi\)
\(37\)	−1.86303	−	9.36607i	−0.306280	−	1.53977i	−0.760770	−	0.649021i	\(-0.775179\pi\)
							0.454491	−	0.890752i	\(-0.349821\pi\)
\(41\)	1.21167	+	9.20357i	0.189232	+	1.43736i	0.777598	+	0.628762i	\(0.216438\pi\)
							−0.588366	+	0.808594i	\(0.700229\pi\)
\(43\)	−9.03235	+	4.45426i	−1.37742	+	0.679269i	−0.972026	−	0.234873i	\(-0.924533\pi\)
							−0.405395	+	0.914142i	\(0.632866\pi\)
\(47\)	−7.40035	+	1.98292i	−1.07945	+	0.289238i	−0.754371	−	0.656448i	\(-0.772058\pi\)
							−0.325080	+	0.945686i	\(0.605391\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.89	yes	1504
9.4	even	3	inner	576.2.bm.a.13.38	✓	1504
64.5	even	16	inner	576.2.bm.a.133.38	yes	1504
576.517	even	48	inner	576.2.bm.a.517.89	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.38	✓	1504	9.4	even	3	inner
576.2.bm.a.133.38	yes	1504	64.5	even	16	inner
576.2.bm.a.205.89	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.89	yes	1504	576.517	even	48	inner