Embedded newform 576.2.bm.a.205.77

• Label: 576.2.bm.a.205.77
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.77
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.77

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.51	✓	1504	9.4	even	3	inner
576.2.bm.a.133.51	yes	1504	64.5	even	16	inner
576.2.bm.a.205.77	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.77	yes	1504	576.517	even	48	inner