Embedded newform 576.2.bl.a.419.6

• Label: 576.2.bl.a.419.6
• Level: $576$
• Weight: $2$
• Character: 576.419
• 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			419.6
Character	\(\chi\)	\(=\)	576.419
Dual form			576.2.bl.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38521 + 0.284957i) q^{2} +(0.799756 - 1.53636i) q^{3} +(1.83760 - 0.789449i) q^{4} +(1.54938 - 0.764070i) q^{5} +(-0.670032 + 2.35607i) q^{6} +(2.14122 - 2.79049i) q^{7} +(-2.32050 + 1.61719i) q^{8} +(-1.72078 - 2.45742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38521	+	0.284957i	−0.979490	+	0.201495i
							\(3\)	0.799756	−	1.53636i	0.461739	−	0.887016i
\(5\)	1.54938	−	0.764070i	0.692904	−	0.341703i								−0.0615050	−	0.998107i	\(-0.519590\pi\)
							0.754409	+	0.656404i	\(0.227923\pi\)
\(7\)	2.14122	−	2.79049i	0.809304	−	1.05471i	−0.188061	−	0.982157i	\(-0.560220\pi\)
							0.997365	−	0.0725479i	\(-0.0231130\pi\)
\(11\)	0.262556	−	0.230255i	0.0791635	−	0.0694245i	−0.618850	−	0.785509i	\(-0.712401\pi\)
							0.698013	+	0.716085i	\(0.254068\pi\)
\(13\)	3.36419	−	0.220501i	0.933060	−	0.0611560i	0.408715	−	0.912662i	\(-0.365977\pi\)
							0.524345	+	0.851506i	\(0.324310\pi\)
\(17\)	−0.546278	+	0.546278i	−0.132492	+	0.132492i	−0.770243	−	0.637751i	\(-0.779865\pi\)
							0.637751	+	0.770243i	\(0.279865\pi\)
\(19\)	−0.385670	−	0.577197i	−0.0884789	−	0.132418i	0.784590	−	0.620015i	\(-0.212874\pi\)
							−0.873069	+	0.487597i	\(0.837874\pi\)
\(23\)	−0.643231	−	0.838275i	−0.134123	−	0.174792i	0.721463	−	0.692453i	\(-0.243470\pi\)
							−0.855586	+	0.517660i	\(0.826803\pi\)
\(29\)	2.99031	+	8.80918i	0.555287	+	1.63582i	0.754820	+	0.655932i	\(0.227724\pi\)
							−0.199533	+	0.979891i	\(0.563942\pi\)
\(31\)	−2.64959	+	4.58922i	−0.475880	+	0.824248i	−0.999618	−	0.0276309i	\(-0.991204\pi\)
							0.523738	+	0.851879i	\(0.324537\pi\)
\(37\)	−1.20183	−	0.803037i	−0.197580	−	0.132019i	0.452845	−	0.891589i	\(-0.350409\pi\)
							−0.650425	+	0.759571i	\(0.725409\pi\)
\(41\)	−2.61841	+	2.00917i	−0.408926	+	0.313780i	−0.792660	−	0.609663i	\(-0.791305\pi\)
							0.383734	+	0.923444i	\(0.374638\pi\)
\(43\)	6.94890	−	6.09402i	1.05970	−	0.929329i	0.0620626	−	0.998072i	\(-0.480232\pi\)
							0.997634	+	0.0687429i	\(0.0218988\pi\)
\(47\)	1.19711	−	4.46769i	0.174617	−	0.651680i	−0.822000	−	0.569488i	\(-0.807141\pi\)
							0.996617	−	0.0821914i	\(-0.0261919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.419.6	yes	1504
9.2	odd	6	inner	576.2.bl.a.227.25	yes	1504
64.11	odd	16	inner	576.2.bl.a.203.25	yes	1504
576.11	even	48	inner	576.2.bl.a.11.6	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.6	✓	1504	576.11	even	48	inner
576.2.bl.a.203.25	yes	1504	64.11	odd	16	inner
576.2.bl.a.227.25	yes	1504	9.2	odd	6	inner
576.2.bl.a.419.6	yes	1504	1.1	even	1	trivial