Embedded newform 576.2.bm.a.205.58

• Label: 576.2.bm.a.205.58
• 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			205.58
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.390119 - 1.35934i) q^{2} +(-1.61102 + 0.636098i) q^{3} +(-1.69561 - 1.06061i) q^{4} +(-2.42602 - 2.76635i) q^{5} +(0.236185 + 2.43808i) q^{6} +(0.616456 + 0.0811579i) q^{7} +(-2.10322 + 1.89115i) q^{8} +(2.19076 - 2.04953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.390119	−	1.35934i	0.275856	−	0.961199i
							\(3\)	−1.61102	+	0.636098i	−0.930122	+	0.367251i
\(5\)	−2.42602	−	2.76635i	−1.08495	−	1.23715i								−0.969827	−	0.243796i	\(-0.921607\pi\)
							−0.115123	−	0.993351i	\(-0.536726\pi\)
\(7\)	0.616456	+	0.0811579i	0.232998	+	0.0306748i	0.246121	−	0.969239i	\(-0.420844\pi\)
							−0.0131228	+	0.999914i	\(0.504177\pi\)
\(11\)	−1.76408	−	3.57720i	−0.531890	−	1.07857i	−0.982572	−	0.185881i	\(-0.940486\pi\)
							0.450683	−	0.892684i	\(-0.351181\pi\)
\(13\)	0.0634404	+	0.186889i	0.0175952	+	0.0518338i	0.955373	−	0.295402i	\(-0.0954536\pi\)
							−0.937778	+	0.347236i	\(0.887120\pi\)
\(17\)	−0.591671	−	0.591671i	−0.143501	−	0.143501i	0.631706	−	0.775208i	\(-0.282355\pi\)
							−0.775208	+	0.631706i	\(0.782355\pi\)
\(19\)	−1.08791	+	5.46928i	−0.249583	+	1.25474i	0.629096	+	0.777327i	\(0.283425\pi\)
							−0.878679	+	0.477412i	\(0.841575\pi\)
\(23\)	1.07209	+	8.14334i	0.223546	+	1.69800i	0.623788	+	0.781594i	\(0.285593\pi\)
							−0.400241	+	0.916410i	\(0.631074\pi\)
\(29\)	−5.68214	+	0.372427i	−1.05515	+	0.0691580i	−0.583088	−	0.812409i	\(-0.698156\pi\)
							−0.472059	+	0.881567i	\(0.656489\pi\)
\(31\)	1.29382	−	0.746987i	0.232377	−	0.134163i	−0.379291	−	0.925277i	\(-0.623832\pi\)
							0.611668	+	0.791114i	\(0.290499\pi\)
\(37\)	−0.255932	−	1.28666i	−0.0420750	−	0.211525i	0.954029	−	0.299715i	\(-0.0968917\pi\)
							−0.996104	+	0.0881901i	\(0.971892\pi\)
\(41\)	−0.984205	−	7.47578i	−0.153707	−	1.16752i	−0.878219	−	0.478258i	\(-0.841268\pi\)
							0.724512	−	0.689262i	\(-0.242065\pi\)
\(43\)	5.78385	−	2.85228i	0.882029	−	0.434969i	0.0558914	−	0.998437i	\(-0.482200\pi\)
							0.826138	+	0.563468i	\(0.190533\pi\)
\(47\)	−0.527824	+	0.141430i	−0.0769910	+	0.0206297i	−0.297109	−	0.954844i	\(-0.596022\pi\)
							0.220118	+	0.975473i	\(0.429356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.58	yes	1504
9.4	even	3	inner	576.2.bm.a.13.70	✓	1504
64.5	even	16	inner	576.2.bm.a.133.70	yes	1504
576.517	even	48	inner	576.2.bm.a.517.58	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.70	✓	1504	9.4	even	3	inner
576.2.bm.a.133.70	yes	1504	64.5	even	16	inner
576.2.bm.a.205.58	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.58	yes	1504	576.517	even	48	inner