Embedded newform 576.2.bm.a.13.4

• Label: 576.2.bm.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40823 - 0.129985i) q^{2} +(1.52000 + 0.830426i) q^{3} +(1.96621 + 0.366096i) q^{4} +(1.17841 - 3.47147i) q^{5} +(-2.03256 - 1.36700i) q^{6} +(0.673012 - 0.877086i) q^{7} +(-2.72128 - 0.771123i) q^{8} +(1.62079 + 2.52449i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40823	−	0.129985i	−0.995767	−	0.0919130i
							\(3\)	1.52000	+	0.830426i	0.877571	+	0.479447i
\(5\)	1.17841	−	3.47147i	0.526999	−	1.55249i								−0.278829	−	0.960341i	\(-0.589946\pi\)
							0.805828	−	0.592150i	\(-0.201721\pi\)
\(7\)	0.673012	−	0.877086i	0.254375	−	0.331507i	−0.648658	−	0.761080i	\(-0.724669\pi\)
							0.903032	+	0.429573i	\(0.141336\pi\)
\(11\)	−1.90239	−	0.124689i	−0.573593	−	0.0375953i	−0.224153	−	0.974554i	\(-0.571962\pi\)
							−0.349440	+	0.936959i	\(0.613628\pi\)
\(13\)	3.78705	−	4.31830i	1.05034	−	1.19768i	0.0705781	−	0.997506i	\(-0.477516\pi\)
							0.979760	−	0.200175i	\(-0.0641511\pi\)
\(17\)	−2.60631	−	2.60631i	−0.632124	−	0.632124i	0.316476	−	0.948600i	\(-0.397500\pi\)
							−0.948600	+	0.316476i	\(0.897500\pi\)
\(19\)	−0.437356	+	2.19874i	−0.100336	+	0.504424i	0.897634	+	0.440742i	\(0.145285\pi\)
							−0.997970	+	0.0636827i	\(0.979715\pi\)
\(23\)	−0.738112	+	0.566374i	−0.153907	+	0.118097i	−0.682844	−	0.730564i	\(-0.739257\pi\)
							0.528937	+	0.848661i	\(0.322591\pi\)
\(29\)	−0.135139	+	0.274035i	−0.0250947	+	0.0508870i	−0.909059	−	0.416667i	\(-0.863198\pi\)
							0.883964	+	0.467554i	\(0.154865\pi\)
\(31\)	−5.52185	−	3.18804i	−0.991754	−	0.572589i	−0.0859559	−	0.996299i	\(-0.527394\pi\)
							−0.905798	+	0.423710i	\(0.860728\pi\)
\(37\)	1.30760	+	6.57376i	0.214968	+	1.08072i	0.925990	+	0.377548i	\(0.123233\pi\)
							−0.711022	+	0.703170i	\(0.751767\pi\)
\(41\)	8.08007	−	6.20005i	1.26189	−	0.968285i	0.261899	−	0.965095i	\(-0.415651\pi\)
							0.999995	−	0.00319027i	\(-0.00101550\pi\)
\(43\)	−0.275302	+	4.20030i	−0.0419832	+	0.640540i	0.924074	+	0.382213i	\(0.124838\pi\)
							−0.966058	+	0.258327i	\(0.916829\pi\)
\(47\)	0.133997	−	0.500085i	0.0195455	−	0.0729448i	−0.955464	−	0.295107i	\(-0.904645\pi\)
							0.975010	+	0.222162i	\(0.0713114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.4	✓	1504
9.7	even	3	inner	576.2.bm.a.205.61	yes	1504
64.5	even	16	inner	576.2.bm.a.517.61	yes	1504
576.133	even	48	inner	576.2.bm.a.133.4	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.4	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.4	yes	1504	576.133	even	48	inner
576.2.bm.a.205.61	yes	1504	9.7	even	3	inner
576.2.bm.a.517.61	yes	1504	64.5	even	16	inner