Embedded newform 576.2.bm.a.205.72

• Label: 576.2.bm.a.205.72
• 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			205.72
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.72

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06334 + 0.932362i) q^{2} +(1.71119 - 0.267987i) q^{3} +(0.261403 + 1.98284i) q^{4} +(1.65695 + 1.88938i) q^{5} +(2.06945 + 1.31049i) q^{6} +(-3.29025 - 0.433169i) q^{7} +(-1.57077 + 2.35217i) q^{8} +(2.85637 - 0.917154i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06334	+	0.932362i	0.751898	+	0.659279i
							\(3\)	1.71119	−	0.267987i	0.987958	−	0.154722i
\(5\)	1.65695	+	1.88938i	0.741008	+	0.844958i								0.992416	−	0.122923i	\(-0.0392269\pi\)
							−0.251408	+	0.967881i	\(0.580894\pi\)
\(7\)	−3.29025	−	0.433169i	−1.24360	−	0.163723i	−0.520133	−	0.854085i	\(-0.674118\pi\)
							−0.723464	+	0.690363i	\(0.757451\pi\)
\(11\)	−0.868018	−	1.76017i	−0.261717	−	0.530710i	0.725860	−	0.687842i	\(-0.241442\pi\)
							−0.987578	+	0.157132i	\(0.949775\pi\)
\(13\)	0.548806	+	1.61673i	0.152211	+	0.448400i	0.996112	−	0.0881000i	\(-0.0280795\pi\)
							−0.843900	+	0.536500i	\(0.819746\pi\)
\(17\)	3.70935	+	3.70935i	0.899650	+	0.899650i	0.995405	−	0.0957546i	\(-0.0305264\pi\)
							−0.0957546	+	0.995405i	\(0.530526\pi\)
\(19\)	0.117665	−	0.591543i	0.0269943	−	0.135709i	−0.964938	−	0.262478i	\(-0.915460\pi\)
							0.991932	+	0.126768i	\(0.0404604\pi\)
\(23\)	−0.821015	−	6.23623i	−0.171194	−	1.30034i	−0.834088	−	0.551631i	\(-0.814006\pi\)
							0.662895	−	0.748713i	\(-0.269328\pi\)
\(29\)	4.63375	−	0.303712i	0.860466	−	0.0563979i	0.371221	−	0.928544i	\(-0.378939\pi\)
							0.489245	+	0.872147i	\(0.337272\pi\)
\(31\)	−3.91038	+	2.25766i	−0.702325	+	0.405487i	−0.808213	−	0.588891i	\(-0.799565\pi\)
							0.105888	+	0.994378i	\(0.466232\pi\)
\(37\)	−0.460520	−	2.31519i	−0.0757090	−	0.380615i	0.924290	−	0.381690i	\(-0.124658\pi\)
							−0.999999	+	0.00107508i	\(0.999658\pi\)
\(41\)	−1.51824	−	11.5321i	−0.237109	−	1.80102i	−0.533338	−	0.845902i	\(-0.679063\pi\)
							0.296229	−	0.955117i	\(-0.404271\pi\)
\(43\)	7.41084	−	3.65462i	1.13014	−	0.557325i	0.221609	−	0.975136i	\(-0.428869\pi\)
							0.908534	+	0.417811i	\(0.137202\pi\)
\(47\)	−10.2078	+	2.73518i	−1.48897	+	0.398967i	−0.909388	−	0.415949i	\(-0.863450\pi\)
							−0.579579	+	0.814916i	\(0.696783\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.72	yes	1504
9.4	even	3	inner	576.2.bm.a.13.10	✓	1504
64.5	even	16	inner	576.2.bm.a.133.10	yes	1504
576.517	even	48	inner	576.2.bm.a.517.72	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.10	✓	1504	9.4	even	3	inner
576.2.bm.a.133.10	yes	1504	64.5	even	16	inner
576.2.bm.a.205.72	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.72	yes	1504	576.517	even	48	inner