Embedded newform 547.2.o.a.4.10

• Label: 547.2.o.a.4.10
• Level: $547$
• Weight: $2$
• Character: 547.4
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $6480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.o (of order \(273\), degree \(144\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(6480\)
Relative dimension:	\(45\) over \(\Q(\zeta_{273})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{273}]$

Embedding invariants

Embedding label			4.10
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91587 - 0.0220482i) q^{2} +(-0.216762 + 0.949696i) q^{3} +(1.67061 + 0.0384563i) q^{4} +(-0.340710 + 0.640190i) q^{5} +(0.436227 - 1.81472i) q^{6} +(1.64679 + 4.05800i) q^{7} +(0.629896 + 0.0217545i) q^{8} +(1.84797 + 0.889935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{273}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91587	−	0.0220482i	−1.35473	−	0.0155904i	−0.668898	−	0.743354i	\(-0.733234\pi\)
							−0.685828	+	0.727764i	\(0.740560\pi\)
\(3\)	−0.216762	+	0.949696i	−0.125148	+	0.548307i	0.873014	+	0.487696i	\(0.162162\pi\)
							−0.998161	+	0.0606120i	\(0.980695\pi\)
\(5\)	−0.340710	+	0.640190i	−0.152370	+	0.286301i	−0.947268	−	0.320441i	\(-0.896169\pi\)
							0.794898	+	0.606743i	\(0.207524\pi\)
\(7\)	1.64679	+	4.05800i	0.622427	+	1.53378i	0.831292	+	0.555836i	\(0.187602\pi\)
							−0.208865	+	0.977944i	\(0.566977\pi\)
\(11\)	1.49705	+	1.55859i	0.451377	+	0.469933i	0.907541	−	0.419962i	\(-0.137957\pi\)
							−0.456165	+	0.889895i	\(0.650777\pi\)
\(13\)	−1.12309	−	0.169279i	−0.311490	−	0.0469496i	−0.00856301	−	0.999963i	\(-0.502726\pi\)
							−0.302927	+	0.953014i	\(0.597964\pi\)
\(17\)	0.943035	−	5.59907i	0.228720	−	1.35797i	−0.600757	−	0.799432i	\(-0.705134\pi\)
							0.829477	−	0.558542i	\(-0.188639\pi\)
\(19\)	2.48078	−	1.33865i	0.569129	−	0.307107i	−0.164806	−	0.986326i	\(-0.552700\pi\)
							0.733935	+	0.679219i	\(0.237681\pi\)
\(23\)	6.52673	+	1.52925i	1.36092	+	0.318870i	0.843622	−	0.536937i	\(-0.180419\pi\)
							0.517295	+	0.855807i	\(0.326939\pi\)
\(29\)	1.44663	+	0.824150i	0.268632	+	0.153041i	0.623375	−	0.781923i	\(-0.285761\pi\)
							−0.354743	+	0.934964i	\(0.615431\pi\)
\(31\)	−3.03233	−	1.09052i	−0.544623	−	0.195864i	0.0516217	−	0.998667i	\(-0.483561\pi\)
							−0.596245	+	0.802803i	\(0.703341\pi\)
\(37\)	−4.13467	+	3.33639i	−0.679735	+	0.548499i	−0.904796	−	0.425846i	\(-0.859976\pi\)
							0.225061	+	0.974345i	\(0.427742\pi\)
\(41\)	−3.07626	−	5.32824i	−0.480431	−	0.832131i	0.519317	−	0.854582i	\(-0.326186\pi\)
							−0.999748	+	0.0224510i	\(0.992853\pi\)
\(43\)	−0.265658	+	2.70725i	−0.0405125	+	0.412852i	0.953320	+	0.301961i	\(0.0976414\pi\)
							−0.993833	+	0.110891i	\(0.964630\pi\)
\(47\)	6.74102	−	0.544192i	0.983279	−	0.0793785i	0.421642	−	0.906762i	\(-0.361454\pi\)
							0.561637	+	0.827384i	\(0.310172\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.10	✓	6480
547.137	even	273	inner	547.2.o.a.137.10	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.10	✓	6480	1.1	even	1	trivial
547.2.o.a.137.10	yes	6480	547.137	even	273	inner