Embedded newform 576.2.bm.a.517.55

• Label: 576.2.bm.a.517.55
• 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			517.55
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.345686 + 1.37131i) q^{2} +(-0.245867 - 1.71451i) q^{3} +(-1.76100 + 0.948089i) q^{4} +(0.286709 - 0.326929i) q^{5} +(2.26614 - 0.929844i) q^{6} +(-0.859540 + 0.113161i) q^{7} +(-1.90888 - 2.08714i) q^{8} +(-2.87910 + 0.843083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.345686	+	1.37131i	0.244437	+	0.969665i
							\(3\)	−0.245867	−	1.71451i	−0.141951	−	0.989874i
\(5\)	0.286709	−	0.326929i	0.128220	−	0.146207i								−0.684167	−	0.729325i	\(-0.739834\pi\)
							0.812387	+	0.583118i	\(0.198168\pi\)
\(7\)	−0.859540	+	0.113161i	−0.324875	+	0.0427707i	−0.291200	−	0.956662i	\(-0.594055\pi\)
							−0.0336753	+	0.999433i	\(0.510721\pi\)
\(11\)	−2.07234	+	4.20228i	−0.624833	+	1.26704i	0.322434	+	0.946592i	\(0.395499\pi\)
							−0.947267	+	0.320444i	\(0.896168\pi\)
\(13\)	−1.47296	+	4.33920i	−0.408525	+	1.20348i	0.526893	+	0.849932i	\(0.323357\pi\)
							−0.935418	+	0.353544i	\(0.884976\pi\)
\(17\)	2.54210	−	2.54210i	0.616549	−	0.616549i	−0.328095	−	0.944645i	\(-0.606407\pi\)
							0.944645	+	0.328095i	\(0.106407\pi\)
\(19\)	0.422578	+	2.12444i	0.0969461	+	0.487381i	0.998502	+	0.0547181i	\(0.0174260\pi\)
							−0.901556	+	0.432663i	\(0.857574\pi\)
\(23\)	−1.19003	+	9.03919i	−0.248139	+	1.88480i	0.183898	+	0.982945i	\(0.441129\pi\)
							−0.432036	+	0.901856i	\(0.642205\pi\)
\(29\)	3.70220	+	0.242655i	0.687481	+	0.0450599i	0.405141	−	0.914254i	\(-0.367223\pi\)
							0.282340	+	0.959314i	\(0.408889\pi\)
\(31\)	−7.63171	−	4.40617i	−1.37070	−	0.791372i	−0.379680	−	0.925118i	\(-0.623966\pi\)
							−0.991016	+	0.133746i	\(0.957299\pi\)
\(37\)	1.03248	−	5.19062i	0.169738	−	0.853332i	−0.798248	−	0.602329i	\(-0.794240\pi\)
							0.967986	−	0.251003i	\(-0.0807605\pi\)
\(41\)	−0.367026	+	2.78784i	−0.0573198	+	0.435387i	0.938590	+	0.345035i	\(0.112133\pi\)
							−0.995910	+	0.0903528i	\(0.971201\pi\)
\(43\)	−3.66588	−	1.80781i	−0.559041	−	0.275689i	0.140739	−	0.990047i	\(-0.455052\pi\)
							−0.699780	+	0.714358i	\(0.746719\pi\)
\(47\)	−11.0865	−	2.97061i	−1.61713	−	0.433309i	−0.666973	−	0.745082i	\(-0.732410\pi\)
							−0.950157	+	0.311773i	\(0.899077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.55	yes	1504
9.7	even	3	inner	576.2.bm.a.133.74	yes	1504
64.13	even	16	inner	576.2.bm.a.13.74	✓	1504
576.205	even	48	inner	576.2.bm.a.205.55	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.74	✓	1504	64.13	even	16	inner
576.2.bm.a.133.74	yes	1504	9.7	even	3	inner
576.2.bm.a.205.55	yes	1504	576.205	even	48	inner
576.2.bm.a.517.55	yes	1504	1.1	even	1	trivial