Embedded newform 576.2.bm.a.205.6

• Label: 576.2.bm.a.205.6
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.6
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.6

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.67	✓	1504	9.4	even	3	inner
576.2.bm.a.133.67	yes	1504	64.5	even	16	inner
576.2.bm.a.205.6	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.6	yes	1504	576.517	even	48	inner