Embedded newform 576.2.bm.a.517.37

• Label: 576.2.bm.a.517.37
• 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			517.37
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.536807 - 1.30837i) q^{2} +(1.62674 - 0.594749i) q^{3} +(-1.42368 + 1.40469i) q^{4} +(-1.11981 + 1.27690i) q^{5} +(-1.65140 - 1.80911i) q^{6} +(-2.38568 + 0.314080i) q^{7} +(2.60209 + 1.10865i) q^{8} +(2.29255 - 1.93500i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.536807	−	1.30837i	−0.379580	−	0.925159i
							\(3\)	1.62674	−	0.594749i	0.939197	−	0.343379i
\(5\)	−1.11981	+	1.27690i	−0.500795	+	0.571047i								−0.945911	−	0.324425i	\(-0.894829\pi\)
							0.445116	+	0.895473i	\(0.353162\pi\)
\(7\)	−2.38568	+	0.314080i	−0.901700	+	0.118711i	−0.567100	−	0.823649i	\(-0.691935\pi\)
							−0.334600	+	0.942360i	\(0.608601\pi\)
\(11\)	−2.80867	+	5.69543i	−0.846847	+	1.71724i	−0.164279	+	0.986414i	\(0.552530\pi\)
							−0.682568	+	0.730822i	\(0.739137\pi\)
\(13\)	−0.870768	+	2.56520i	−0.241508	+	0.711459i	0.756819	+	0.653625i	\(0.226753\pi\)
							−0.998326	+	0.0578336i	\(0.981581\pi\)
\(17\)	2.76608	−	2.76608i	0.670874	−	0.670874i	−0.287044	−	0.957917i	\(-0.592673\pi\)
							0.957917	+	0.287044i	\(0.0926725\pi\)
\(19\)	1.19129	+	5.98901i	0.273300	+	1.37397i	0.836643	+	0.547749i	\(0.184515\pi\)
							−0.563342	+	0.826224i	\(0.690485\pi\)
\(23\)	0.265750	−	2.01857i	0.0554128	−	0.420902i	−0.941145	−	0.338003i	\(-0.890249\pi\)
							0.996558	−	0.0828992i	\(-0.0264180\pi\)
\(29\)	−7.63523	−	0.500439i	−1.41783	−	0.0929293i	−0.662885	−	0.748721i	\(-0.730668\pi\)
							−0.754942	+	0.655792i	\(0.772335\pi\)
\(31\)	6.62077	+	3.82250i	1.18912	+	0.686541i	0.958108	−	0.286408i	\(-0.0924614\pi\)
							0.231017	+	0.972950i	\(0.425795\pi\)
\(37\)	−1.71391	+	8.61640i	−0.281765	+	1.41653i	0.537574	+	0.843217i	\(0.319341\pi\)
							−0.819339	+	0.573310i	\(0.805659\pi\)
\(41\)	0.539925	−	4.10114i	0.0843221	−	0.640490i	−0.896138	−	0.443775i	\(-0.853639\pi\)
							0.980461	−	0.196716i	\(-0.0630276\pi\)
\(43\)	5.33531	+	2.63108i	0.813627	+	0.401236i	0.800986	−	0.598684i	\(-0.204309\pi\)
							0.0126415	+	0.999920i	\(0.495976\pi\)
\(47\)	−4.40576	−	1.18052i	−0.642646	−	0.172196i	−0.0772444	−	0.997012i	\(-0.524612\pi\)
							−0.565402	+	0.824816i	\(0.691279\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.37	yes	1504
9.7	even	3	inner	576.2.bm.a.133.27	yes	1504
64.13	even	16	inner	576.2.bm.a.13.27	✓	1504
576.205	even	48	inner	576.2.bm.a.205.37	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.27	✓	1504	64.13	even	16	inner
576.2.bm.a.133.27	yes	1504	9.7	even	3	inner
576.2.bm.a.205.37	yes	1504	576.205	even	48	inner
576.2.bm.a.517.37	yes	1504	1.1	even	1	trivial