Embedded newform 576.2.bm.a.13.67

• Label: 576.2.bm.a.13.67
• 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			13.67
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.67

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.886976 - 1.10149i) q^{2} +(-1.68560 - 0.398433i) q^{3} +(-0.426547 - 1.95399i) q^{4} +(-1.10919 + 3.26757i) q^{5} +(-1.93396 + 1.50327i) q^{6} +(-0.175045 + 0.228123i) q^{7} +(-2.53063 - 1.26330i) q^{8} +(2.68250 + 1.34320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.886976	−	1.10149i	0.627187	−	0.778869i
							\(3\)	−1.68560	−	0.398433i	−0.973182	−	0.230036i
\(5\)	−1.10919	+	3.26757i	−0.496044	+	1.46130i								0.354329	+	0.935121i	\(0.384709\pi\)
							−0.850374	+	0.526179i	\(0.823624\pi\)
\(7\)	−0.175045	+	0.228123i	−0.0661606	+	0.0862222i	−0.825261	−	0.564751i	\(-0.808972\pi\)
							0.759101	+	0.650973i	\(0.225639\pi\)
\(11\)	2.65942	+	0.174308i	0.801847	+	0.0525558i	0.460815	−	0.887496i	\(-0.347557\pi\)
							0.341031	+	0.940052i	\(0.389224\pi\)
\(13\)	3.46634	−	3.95261i	0.961391	−	1.09626i	−0.0339099	−	0.999425i	\(-0.510796\pi\)
							0.995301	−	0.0968313i	\(-0.0308707\pi\)
\(17\)	3.89659	+	3.89659i	0.945061	+	0.945061i	0.998568	−	0.0535062i	\(-0.0170397\pi\)
							−0.0535062	+	0.998568i	\(0.517040\pi\)
\(19\)	0.919464	−	4.62246i	0.210940	−	1.06046i	−0.719632	−	0.694355i	\(-0.755690\pi\)
							0.930572	−	0.366109i	\(-0.119310\pi\)
\(23\)	5.09505	−	3.90957i	1.06239	−	0.815201i	0.0790316	−	0.996872i	\(-0.474817\pi\)
							0.983359	+	0.181671i	\(0.0581505\pi\)
\(29\)	−3.67939	+	7.46106i	−0.683245	+	1.38548i	0.228017	+	0.973657i	\(0.426776\pi\)
							−0.911262	+	0.411827i	\(0.864891\pi\)
\(31\)	4.64561	+	2.68214i	0.834376	+	0.481727i	0.855349	−	0.518053i	\(-0.173343\pi\)
							−0.0209726	+	0.999780i	\(0.506676\pi\)
\(37\)	0.680158	+	3.41939i	0.111817	+	0.562144i	0.995557	+	0.0941641i	\(0.0300179\pi\)
							−0.883739	+	0.467979i	\(0.844982\pi\)
\(41\)	9.50555	−	7.29387i	1.48452	−	1.13911i	0.527323	−	0.849665i	\(-0.323196\pi\)
							0.957194	−	0.289446i	\(-0.0934711\pi\)
\(43\)	−0.586483	+	8.94801i	−0.0894379	+	1.36456i	0.683442	+	0.730005i	\(0.260482\pi\)
							−0.772879	+	0.634553i	\(0.781184\pi\)
\(47\)	0.808685	−	3.01805i	0.117959	−	0.440228i	−0.881532	−	0.472123i	\(-0.843488\pi\)
							0.999491	+	0.0318953i	\(0.0101543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.67	✓	1504
9.7	even	3	inner	576.2.bm.a.205.6	yes	1504
64.5	even	16	inner	576.2.bm.a.517.6	yes	1504
576.133	even	48	inner	576.2.bm.a.133.67	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.67	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.67	yes	1504	576.133	even	48	inner
576.2.bm.a.205.6	yes	1504	9.7	even	3	inner
576.2.bm.a.517.6	yes	1504	64.5	even	16	inner