Embedded newform 576.2.bm.a.13.54

• Label: 576.2.bm.a.13.54
• 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			13.54
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.274719 + 1.38727i) q^{2} +(1.54696 - 0.779045i) q^{3} +(-1.84906 + 0.762222i) q^{4} +(1.10806 - 3.26423i) q^{5} +(1.50573 + 1.93204i) q^{6} +(0.985297 - 1.28406i) q^{7} +(-1.56538 - 2.35575i) q^{8} +(1.78618 - 2.41030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.274719	+	1.38727i	0.194256	+	0.980951i
							\(3\)	1.54696	−	0.779045i	0.893139	−	0.449782i
\(5\)	1.10806	−	3.26423i	0.495539	−	1.45981i								−0.355485	−	0.934682i	\(-0.615684\pi\)
							0.851024	−	0.525127i	\(-0.175982\pi\)
\(7\)	0.985297	−	1.28406i	0.372407	−	0.485331i	−0.569126	−	0.822250i	\(-0.692718\pi\)
							0.941533	+	0.336920i	\(0.109385\pi\)
\(11\)	−4.06980	−	0.266749i	−1.22709	−	0.0804278i	−0.561966	−	0.827160i	\(-0.689955\pi\)
							−0.665125	+	0.746732i	\(0.731622\pi\)
\(13\)	−1.99751	+	2.27773i	−0.554011	+	0.631728i	−0.959276	−	0.282471i	\(-0.908846\pi\)
							0.405265	+	0.914199i	\(0.367179\pi\)
\(17\)	1.07915	+	1.07915i	0.261733	+	0.261733i	0.825758	−	0.564025i	\(-0.190748\pi\)
							−0.564025	+	0.825758i	\(0.690748\pi\)
\(19\)	1.23163	−	6.19184i	0.282556	−	1.42051i	−0.535095	−	0.844792i	\(-0.679724\pi\)
							0.817651	−	0.575714i	\(-0.195276\pi\)
\(23\)	0.208524	−	0.160006i	0.0434803	−	0.0333636i	−0.586793	−	0.809737i	\(-0.699610\pi\)
							0.630273	+	0.776374i	\(0.282943\pi\)
\(29\)	−0.261840	+	0.530959i	−0.0486225	+	0.0985966i	−0.919828	−	0.392322i	\(-0.871672\pi\)
							0.871206	+	0.490918i	\(0.163339\pi\)
\(31\)	6.42245	+	3.70801i	1.15351	+	0.665978i	0.949739	−	0.313042i	\(-0.101348\pi\)
							0.203768	+	0.979019i	\(0.434681\pi\)
\(37\)	2.25001	+	11.3115i	0.369899	+	1.85961i	0.497048	+	0.867723i	\(0.334417\pi\)
							−0.127149	+	0.991884i	\(0.540583\pi\)
\(41\)	9.72083	−	7.45905i	1.51814	−	1.16491i	0.582045	−	0.813156i	\(-0.302253\pi\)
							0.936093	−	0.351752i	\(-0.114414\pi\)
\(43\)	−0.406521	+	6.20231i	−0.0619938	+	0.945843i	0.847677	+	0.530513i	\(0.178001\pi\)
							−0.909671	+	0.415330i	\(0.863666\pi\)
\(47\)	0.799230	−	2.98277i	0.116580	−	0.435081i	−0.882821	−	0.469710i	\(-0.844358\pi\)
							0.999400	+	0.0346292i	\(0.0110250\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.54	✓	1504
9.7	even	3	inner	576.2.bm.a.205.73	yes	1504
64.5	even	16	inner	576.2.bm.a.517.73	yes	1504
576.133	even	48	inner	576.2.bm.a.133.54	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.54	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.54	yes	1504	576.133	even	48	inner
576.2.bm.a.205.73	yes	1504	9.7	even	3	inner
576.2.bm.a.517.73	yes	1504	64.5	even	16	inner