Embedded newform 576.2.bl.a.11.20

• Label: 576.2.bl.a.11.20
• Level: $576$
• Weight: $2$
• Character: 576.11
• 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.bl (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			11.20
Character	\(\chi\)	\(=\)	576.11
Dual form			576.2.bl.a.419.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11718 - 0.867123i) q^{2} +(1.58760 + 0.692470i) q^{3} +(0.496195 + 1.93747i) q^{4} +(0.901826 + 0.444731i) q^{5} +(-1.17319 - 2.15026i) q^{6} +(-0.914237 - 1.19146i) q^{7} +(1.12569 - 2.59477i) q^{8} +(2.04097 + 2.19873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11718	−	0.867123i	−0.789968	−	0.613149i
							\(3\)	1.58760	+	0.692470i	0.916603	+	0.399798i
\(5\)	0.901826	+	0.444731i	0.403309	+	0.198890i								0.632618	−	0.774464i	\(-0.281980\pi\)
							−0.229310	+	0.973354i	\(0.573647\pi\)
\(7\)	−0.914237	−	1.19146i	−0.345549	−	0.450329i	0.587956	−	0.808893i	\(-0.299933\pi\)
							−0.933505	+	0.358565i	\(0.883266\pi\)
\(11\)	2.65213	+	2.32585i	0.799646	+	0.701271i	0.958206	−	0.286079i	\(-0.0923519\pi\)
							−0.158560	+	0.987349i	\(0.550685\pi\)
\(13\)	−0.728842	−	0.0477708i	−0.202144	−	0.0132492i	−0.0360054	−	0.999352i	\(-0.511463\pi\)
							−0.166139	+	0.986102i	\(0.553130\pi\)
\(17\)	1.82931	+	1.82931i	0.443672	+	0.443672i	0.893244	−	0.449572i	\(-0.148423\pi\)
							−0.449572	+	0.893244i	\(0.648423\pi\)
\(19\)	4.23593	−	6.33951i	0.971788	−	1.45438i	0.0827393	−	0.996571i	\(-0.473633\pi\)
							0.889049	−	0.457812i	\(-0.151367\pi\)
\(23\)	−1.44630	+	1.88486i	−0.301575	+	0.393020i	−0.919418	−	0.393281i	\(-0.871340\pi\)
							0.617843	+	0.786301i	\(0.288007\pi\)
\(29\)	−1.63080	+	4.80418i	−0.302832	+	0.892114i	0.683874	+	0.729600i	\(0.260294\pi\)
							−0.986706	+	0.162514i	\(0.948040\pi\)
\(31\)	3.23040	+	5.59521i	0.580196	+	1.00493i	0.995456	+	0.0952265i	\(0.0303575\pi\)
							−0.415259	+	0.909703i	\(0.636309\pi\)
\(37\)	−2.15421	+	1.43940i	−0.354150	+	0.236635i	−0.719900	−	0.694078i	\(-0.755812\pi\)
							0.365750	+	0.930713i	\(0.380812\pi\)
\(41\)	6.09545	+	4.67720i	0.951949	+	0.730456i	0.963005	−	0.269483i	\(-0.0868528\pi\)
							−0.0110565	+	0.999939i	\(0.503519\pi\)
\(43\)	3.03543	+	2.66200i	0.462899	+	0.405952i	0.858775	−	0.512352i	\(-0.171226\pi\)
							−0.395876	+	0.918304i	\(0.629559\pi\)
\(47\)	0.218232	+	0.814451i	0.0318324	+	0.118800i	0.980014	−	0.198929i	\(-0.0637464\pi\)
							−0.948182	+	0.317729i	\(0.897080\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.11.20	✓	1504
9.5	odd	6	inner	576.2.bl.a.203.13	yes	1504
64.35	odd	16	inner	576.2.bl.a.227.13	yes	1504
576.419	even	48	inner	576.2.bl.a.419.20	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.20	✓	1504	1.1	even	1	trivial
576.2.bl.a.203.13	yes	1504	9.5	odd	6	inner
576.2.bl.a.227.13	yes	1504	64.35	odd	16	inner
576.2.bl.a.419.20	yes	1504	576.419	even	48	inner