Embedded newform 576.2.bm.a.517.6

• Label: 576.2.bm.a.517.6
• 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			517.6
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39740 + 0.217400i) q^{2} +(-0.276947 - 1.70977i) q^{3} +(1.90547 - 0.607592i) q^{4} +(-2.27520 + 2.59437i) q^{5} +(0.758710 + 2.32903i) q^{6} +(0.285082 - 0.0375318i) q^{7} +(-2.53063 + 1.26330i) q^{8} +(-2.84660 + 0.947029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39740	+	0.217400i	−0.988114	+	0.153725i
							\(3\)	−0.276947	−	1.70977i	−0.159895	−	0.987134i
\(5\)	−2.27520	+	2.59437i	−1.01750	+	1.16024i								−0.0304975	+	0.999535i	\(0.509709\pi\)
							−0.987003	+	0.160702i	\(0.948624\pi\)
\(7\)	0.285082	−	0.0375318i	0.107751	−	0.0141857i	−0.0764579	−	0.997073i	\(-0.524361\pi\)
							0.184209	+	0.982887i	\(0.441028\pi\)
\(11\)	−1.17876	+	2.39028i	−0.355409	+	0.720698i	−0.999002	−	0.0446703i	\(-0.985776\pi\)
							0.643593	+	0.765368i	\(0.277443\pi\)
\(13\)	1.68989	−	4.97825i	0.468690	−	1.38072i	−0.413792	−	0.910371i	\(-0.635796\pi\)
							0.882482	−	0.470346i	\(-0.155871\pi\)
\(17\)	3.89659	−	3.89659i	0.945061	−	0.945061i	−0.0535062	−	0.998568i	\(-0.517040\pi\)
							0.998568	+	0.0535062i	\(0.0170397\pi\)
\(19\)	0.919464	+	4.62246i	0.210940	+	1.06046i	0.930572	+	0.366109i	\(0.119310\pi\)
							−0.719632	+	0.694355i	\(0.755690\pi\)
\(23\)	0.838261	−	6.36723i	0.174790	−	1.32766i	−0.649011	−	0.760779i	\(-0.724817\pi\)
							0.823801	−	0.566879i	\(-0.191849\pi\)
\(29\)	8.30116	+	0.544087i	1.54149	+	0.101034i	0.812283	−	0.583263i	\(-0.198224\pi\)
							0.729203	+	0.684297i	\(0.239891\pi\)
\(31\)	−4.64561	−	2.68214i	−0.834376	−	0.481727i	0.0209726	−	0.999780i	\(-0.493324\pi\)
							−0.855349	+	0.518053i	\(0.826657\pi\)
\(37\)	0.680158	−	3.41939i	0.111817	−	0.562144i	−0.883739	−	0.467979i	\(-0.844982\pi\)
							0.995557	−	0.0941641i	\(-0.0300179\pi\)
\(41\)	1.56390	−	11.8790i	0.244240	−	1.85519i	−0.227930	−	0.973678i	\(-0.573196\pi\)
							0.472169	−	0.881508i	\(-0.343471\pi\)
\(43\)	8.04244	+	3.96609i	1.22646	+	0.604824i	0.935979	−	0.352057i	\(-0.114518\pi\)
							0.290482	+	0.956880i	\(0.406184\pi\)
\(47\)	−3.01805	−	0.808685i	−0.440228	−	0.117959i	0.0318953	−	0.999491i	\(-0.489846\pi\)
							−0.472123	+	0.881532i	\(0.656512\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.67	✓	1504	64.13	even	16	inner
576.2.bm.a.133.67	yes	1504	9.7	even	3	inner
576.2.bm.a.205.6	yes	1504	576.205	even	48	inner
576.2.bm.a.517.6	yes	1504	1.1	even	1	trivial