Embedded newform 576.2.bm.a.13.27

• Label: 576.2.bm.a.13.27
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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			13.27
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.864680 - 1.11907i) q^{2} +(0.0730485 - 1.73051i) q^{3} +(-0.504657 + 1.93528i) q^{4} +(-0.545923 + 1.60824i) q^{5} +(-1.99973 + 1.41459i) q^{6} +(1.46484 - 1.90902i) q^{7} +(2.60209 - 1.10865i) q^{8} +(-2.98933 - 0.252822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.864680	−	1.11907i	−0.611421	−	0.791305i
							\(3\)	0.0730485	−	1.73051i	0.0421746	−	0.999110i
\(5\)	−0.545923	+	1.60824i	−0.244144	+	0.719225i								0.753916	+	0.656970i	\(0.228162\pi\)
							−0.998060	+	0.0622545i	\(0.980171\pi\)
\(7\)	1.46484	−	1.90902i	0.553657	−	0.721540i	−0.429751	−	0.902947i	\(-0.641399\pi\)
							0.983408	+	0.181408i	\(0.0580653\pi\)
\(11\)	6.33672	+	0.415331i	1.91059	+	0.125227i	0.974195	−	0.225710i	\(-0.0724701\pi\)
							0.936399	+	0.350937i	\(0.114137\pi\)
\(13\)	−1.78614	+	2.03671i	−0.495387	+	0.564881i	−0.944465	−	0.328612i	\(-0.893419\pi\)
							0.449078	+	0.893493i	\(0.351753\pi\)
\(17\)	2.76608	+	2.76608i	0.670874	+	0.670874i	0.957917	−	0.287044i	\(-0.0926725\pi\)
							−0.287044	+	0.957917i	\(0.592673\pi\)
\(19\)	1.19129	−	5.98901i	0.273300	−	1.37397i	−0.563342	−	0.826224i	\(-0.690485\pi\)
							0.836643	−	0.547749i	\(-0.184515\pi\)
\(23\)	1.61526	−	1.23943i	0.336805	−	0.258440i	−0.426486	−	0.904494i	\(-0.640249\pi\)
							0.763291	+	0.646054i	\(0.223582\pi\)
\(29\)	3.38422	−	6.86252i	0.628434	−	1.27434i	−0.316969	−	0.948436i	\(-0.602665\pi\)
							0.945403	−	0.325903i	\(-0.105668\pi\)
\(31\)	−6.62077	−	3.82250i	−1.18912	−	0.686541i	−0.231017	−	0.972950i	\(-0.574205\pi\)
							−0.958108	+	0.286408i	\(0.907539\pi\)
\(37\)	−1.71391	−	8.61640i	−0.281765	−	1.41653i	−0.819339	−	0.573310i	\(-0.805659\pi\)
							0.537574	−	0.843217i	\(-0.319341\pi\)
\(41\)	3.28173	−	2.51816i	0.512520	−	0.393270i	−0.319870	−	0.947462i	\(-0.603639\pi\)
							0.832389	+	0.554191i	\(0.186972\pi\)
\(43\)	−0.389070	+	5.93606i	−0.0593326	+	0.905240i	0.859635	+	0.510908i	\(0.170691\pi\)
							−0.918968	+	0.394332i	\(0.870976\pi\)
\(47\)	1.18052	−	4.40576i	0.172196	−	0.642646i	−0.824816	−	0.565402i	\(-0.808721\pi\)
							0.997012	−	0.0772444i	\(-0.0246122\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.27	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.27	yes	1504	576.133	even	48	inner
576.2.bm.a.205.37	yes	1504	9.7	even	3	inner
576.2.bm.a.517.37	yes	1504	64.5	even	16	inner