Embedded newform 576.2.bm.a.517.74

• Label: 576.2.bm.a.517.74
• 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			517.74
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.74

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07853 + 0.914752i) q^{2} +(-1.71952 - 0.208009i) q^{3} +(0.326458 + 1.97318i) q^{4} +(1.32420 - 1.50996i) q^{5} +(-1.66427 - 1.79727i) q^{6} +(-3.08898 + 0.406671i) q^{7} +(-1.45287 + 2.42676i) q^{8} +(2.91346 + 0.715350i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07853	+	0.914752i	0.762636	+	0.646827i
							\(3\)	−1.71952	−	0.208009i	−0.992763	−	0.120094i
\(5\)	1.32420	−	1.50996i	0.592200	−	0.675275i								−0.375662	−	0.926757i	\(-0.622585\pi\)
							0.967862	+	0.251482i	\(0.0809179\pi\)
\(7\)	−3.08898	+	0.406671i	−1.16752	+	0.153707i	−0.689263	−	0.724511i	\(-0.742066\pi\)
							−0.478260	+	0.878218i	\(0.658732\pi\)
\(11\)	−2.05267	+	4.16240i	−0.618903	+	1.25501i	0.331343	+	0.943510i	\(0.392498\pi\)
							−0.950246	+	0.311501i	\(0.899168\pi\)
\(13\)	0.0196135	−	0.0577796i	0.00543982	−	0.0160252i	−0.944176	−	0.329441i	\(-0.893140\pi\)
							0.949616	+	0.313415i	\(0.101473\pi\)
\(17\)	−2.39965	+	2.39965i	−0.582002	+	0.582002i	−0.935453	−	0.353451i	\(-0.885008\pi\)
							0.353451	+	0.935453i	\(0.385008\pi\)
\(19\)	0.881841	+	4.43331i	0.202308	+	1.01707i	0.939802	+	0.341718i	\(0.111009\pi\)
							−0.737494	+	0.675353i	\(0.763991\pi\)
\(23\)	−0.0159916	+	0.121468i	−0.00333447	+	0.0253278i	−0.993031	−	0.117852i	\(-0.962399\pi\)
							0.989697	+	0.143179i	\(0.0457326\pi\)
\(29\)	−1.97603	−	0.129516i	−0.366939	−	0.0240505i	−0.119181	−	0.992872i	\(-0.538027\pi\)
							−0.247758	+	0.968822i	\(0.579694\pi\)
\(31\)	1.81908	+	1.05024i	0.326716	+	0.188630i	0.654382	−	0.756164i	\(-0.272929\pi\)
							−0.327666	+	0.944794i	\(0.606262\pi\)
\(37\)	0.634863	−	3.19167i	0.104371	−	0.524708i	−0.892860	−	0.450335i	\(-0.851305\pi\)
							0.997231	−	0.0743726i	\(-0.0236954\pi\)
\(41\)	−0.752383	+	5.71491i	−0.117502	+	0.892519i	0.826582	+	0.562816i	\(0.190282\pi\)
							−0.944085	+	0.329703i	\(0.893051\pi\)
\(43\)	−6.77492	−	3.34102i	−1.03317	−	0.509501i	−0.155029	−	0.987910i	\(-0.549547\pi\)
							−0.878137	+	0.478409i	\(0.841214\pi\)
\(47\)	8.22030	+	2.20262i	1.19905	+	0.321285i	0.802458	−	0.596708i	\(-0.203525\pi\)
							0.396595	+	0.917994i	\(0.370192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.74	yes	1504
9.7	even	3	inner	576.2.bm.a.133.53	yes	1504
64.13	even	16	inner	576.2.bm.a.13.53	✓	1504
576.205	even	48	inner	576.2.bm.a.205.74	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.53	✓	1504	64.13	even	16	inner
576.2.bm.a.133.53	yes	1504	9.7	even	3	inner
576.2.bm.a.205.74	yes	1504	576.205	even	48	inner
576.2.bm.a.517.74	yes	1504	1.1	even	1	trivial