Embedded newform 576.2.bm.a.517.77

• Label: 576.2.bm.a.517.77
• 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			517.77
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.77

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.11969 + 0.863881i) q^{2} +(-1.07977 - 1.35429i) q^{3} +(0.507418 + 1.93456i) q^{4} +(-1.03601 + 1.18134i) q^{5} +(-0.0390633 - 2.44918i) q^{6} +(3.17505 - 0.418003i) q^{7} +(-1.10308 + 2.60446i) q^{8} +(-0.668196 + 2.92464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.11969	+	0.863881i	0.791741	+	0.610856i
							\(3\)	−1.07977	−	1.35429i	−0.623405	−	0.781899i
\(5\)	−1.03601	+	1.18134i	−0.463317	+	0.528312i								−0.935565	−	0.353154i	\(-0.885109\pi\)
							0.472248	+	0.881466i	\(0.343443\pi\)
\(7\)	3.17505	−	0.418003i	1.20006	−	0.157990i	0.496120	−	0.868254i	\(-0.334758\pi\)
							0.703936	+	0.710264i	\(0.251424\pi\)
\(11\)	1.01361	−	2.05539i	0.305613	−	0.619723i	−0.688838	−	0.724915i	\(-0.741879\pi\)
							0.994452	+	0.105192i	\(0.0335458\pi\)
\(13\)	−1.01576	+	2.99234i	−0.281722	+	0.829927i	0.710304	+	0.703895i	\(0.248557\pi\)
							−0.992026	+	0.126032i	\(0.959776\pi\)
\(17\)	1.39316	−	1.39316i	0.337891	−	0.337891i	−0.517682	−	0.855573i	\(-0.673205\pi\)
							0.855573	+	0.517682i	\(0.173205\pi\)
\(19\)	1.37318	+	6.90343i	0.315028	+	1.58375i	0.736194	+	0.676771i	\(0.236621\pi\)
							−0.421166	+	0.906984i	\(0.638379\pi\)
\(23\)	−0.00698948	+	0.0530904i	−0.00145741	+	0.0110701i	−0.992158	−	0.124989i	\(-0.960110\pi\)
							0.990701	+	0.136059i	\(0.0434437\pi\)
\(29\)	2.66685	+	0.174795i	0.495221	+	0.0324585i	0.310968	−	0.950420i	\(-0.399347\pi\)
							0.184253	+	0.982879i	\(0.441013\pi\)
\(31\)	5.37141	+	3.10119i	0.964735	+	0.556990i	0.897627	−	0.440756i	\(-0.145290\pi\)
							0.0671076	+	0.997746i	\(0.478623\pi\)
\(37\)	−1.25906	+	6.32974i	−0.206989	+	1.04060i	0.727905	+	0.685678i	\(0.240494\pi\)
							−0.934894	+	0.354926i	\(0.884506\pi\)
\(41\)	1.41570	−	10.7533i	0.221095	−	1.67939i	−0.416594	−	0.909093i	\(-0.636776\pi\)
							0.637689	−	0.770294i	\(-0.279890\pi\)
\(43\)	−8.12475	−	4.00668i	−1.23901	−	0.611013i	−0.299686	−	0.954038i	\(-0.596882\pi\)
							−0.939326	+	0.343025i	\(0.888549\pi\)
\(47\)	8.22210	+	2.20311i	1.19932	+	0.321356i	0.802562	−	0.596569i	\(-0.203470\pi\)
							0.396755	+	0.917925i	\(0.370136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.77	yes	1504
9.7	even	3	inner	576.2.bm.a.133.51	yes	1504
64.13	even	16	inner	576.2.bm.a.13.51	✓	1504
576.205	even	48	inner	576.2.bm.a.205.77	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.51	✓	1504	64.13	even	16	inner
576.2.bm.a.133.51	yes	1504	9.7	even	3	inner
576.2.bm.a.205.77	yes	1504	576.205	even	48	inner
576.2.bm.a.517.77	yes	1504	1.1	even	1	trivial