Embedded newform 576.2.bm.a.13.55

• Label: 576.2.bm.a.13.55
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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			13.55
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.323568 + 1.37670i) q^{2} +(1.14915 - 1.29594i) q^{3} +(-1.79061 + 0.890912i) q^{4} +(-0.141063 + 0.415559i) q^{5} +(2.15595 + 1.16271i) q^{6} +(0.0811473 - 0.105753i) q^{7} +(-1.80590 - 2.17686i) q^{8} +(-0.358909 - 2.97845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.323568	+	1.37670i	0.228797	+	0.973474i
							\(3\)	1.14915	−	1.29594i	0.663462	−	0.748210i
\(5\)	−0.141063	+	0.415559i	−0.0630854	+	0.185844i								−0.973903	−	0.226966i	\(-0.927119\pi\)
							0.910817	+	0.412810i	\(0.135453\pi\)
\(7\)	0.0811473	−	0.105753i	0.0306708	−	0.0399709i	−0.777766	−	0.628554i	\(-0.783647\pi\)
							0.808437	+	0.588583i	\(0.200314\pi\)
\(11\)	5.54328	+	0.363326i	1.67136	+	0.109547i	0.871422	−	0.490534i	\(-0.163198\pi\)
							0.799940	+	0.600081i	\(0.204865\pi\)
\(13\)	2.94432	−	3.35736i	0.816608	−	0.931163i	−0.182032	−	0.983293i	\(-0.558267\pi\)
							0.998640	+	0.0521292i	\(0.0166008\pi\)
\(17\)	0.861641	+	0.861641i	0.208979	+	0.208979i	0.803833	−	0.594855i	\(-0.202790\pi\)
							−0.594855	+	0.803833i	\(0.702790\pi\)
\(19\)	−1.11562	+	5.60861i	−0.255941	+	1.28670i	0.612326	+	0.790605i	\(0.290234\pi\)
							−0.868267	+	0.496097i	\(0.834766\pi\)
\(23\)	2.90596	−	2.22982i	0.605935	−	0.464950i	−0.259657	−	0.965701i	\(-0.583609\pi\)
							0.865591	+	0.500751i	\(0.166943\pi\)
\(29\)	−2.28732	+	4.63822i	−0.424744	+	0.861296i	0.574345	+	0.818614i	\(0.305257\pi\)
							−0.999089	+	0.0426823i	\(0.986410\pi\)
\(31\)	−6.68600	−	3.86017i	−1.20084	−	0.693306i	−0.240099	−	0.970748i	\(-0.577180\pi\)
							−0.960742	+	0.277442i	\(0.910513\pi\)
\(37\)	0.738679	+	3.71359i	0.121438	+	0.610510i	0.992791	+	0.119855i	\(0.0382428\pi\)
							−0.871353	+	0.490656i	\(0.836757\pi\)
\(41\)	2.96196	−	2.27279i	0.462580	−	0.354950i	−0.351027	−	0.936365i	\(-0.614167\pi\)
							0.813607	+	0.581415i	\(0.197501\pi\)
\(43\)	0.220134	−	3.35860i	0.0335701	−	0.512181i	−0.947653	−	0.319303i	\(-0.896551\pi\)
							0.981223	−	0.192878i	\(-0.0617822\pi\)
\(47\)	0.429538	−	1.60306i	0.0626545	−	0.233830i	−0.927497	−	0.373831i	\(-0.878044\pi\)
							0.990151	+	0.140001i	\(0.0447106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.55	✓	1504
9.7	even	3	inner	576.2.bm.a.205.71	yes	1504
64.5	even	16	inner	576.2.bm.a.517.71	yes	1504
576.133	even	48	inner	576.2.bm.a.133.55	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.55	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.55	yes	1504	576.133	even	48	inner
576.2.bm.a.205.71	yes	1504	9.7	even	3	inner
576.2.bm.a.517.71	yes	1504	64.5	even	16	inner