Embedded newform 547.2.o.a.4.15

• Label: 547.2.o.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17840 - 0.0135612i) q^{2} +(0.739679 - 3.24075i) q^{3} +(-0.611031 - 0.0140656i) q^{4} +(1.18492 - 2.22646i) q^{5} +(-0.915585 + 3.80886i) q^{6} +(1.56661 + 3.86044i) q^{7} +(3.07540 + 0.106214i) q^{8} +(-7.25240 - 3.49257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17840	−	0.0135612i	−0.833254	−	0.00958923i	−0.406166	−	0.913799i	\(-0.633134\pi\)
							−0.427088	+	0.904210i	\(0.640460\pi\)
\(3\)	0.739679	−	3.24075i	0.427054	−	1.87105i	−0.0608478	−	0.998147i	\(-0.519380\pi\)
							0.487902	−	0.872899i	\(-0.337762\pi\)
\(5\)	1.18492	−	2.22646i	0.529914	−	0.995702i	−0.464030	−	0.885819i	\(-0.653597\pi\)
							0.993945	−	0.109883i	\(-0.0350475\pi\)
\(7\)	1.56661	+	3.86044i	0.592125	+	1.45911i	0.867334	+	0.497726i	\(0.165831\pi\)
							−0.275210	+	0.961384i	\(0.588747\pi\)
\(11\)	−2.31834	−	2.41364i	−0.699005	−	0.727741i	0.273178	−	0.961963i	\(-0.411925\pi\)
							−0.972183	+	0.234222i	\(0.924746\pi\)
\(13\)	−4.55119	−	0.685982i	−1.26227	−	0.190257i	−0.516402	−	0.856346i	\(-0.672729\pi\)
							−0.745872	+	0.666089i	\(0.767967\pi\)
\(17\)	0.601691	−	3.57241i	0.145931	−	0.866436i	−0.812888	−	0.582420i	\(-0.802106\pi\)
							0.958820	−	0.284016i	\(-0.0916669\pi\)
\(19\)	1.13965	−	0.614961i	0.261453	−	0.141082i	−0.338887	−	0.940827i	\(-0.610050\pi\)
							0.600340	+	0.799745i	\(0.295032\pi\)
\(23\)	−3.31305	−	0.776264i	−0.690818	−	0.161862i	−0.132121	−	0.991234i	\(-0.542179\pi\)
							−0.558698	+	0.829371i	\(0.688699\pi\)
\(29\)	−4.28066	−	2.43871i	−0.794898	−	0.452856i	0.0427373	−	0.999086i	\(-0.486392\pi\)
							−0.837635	+	0.546230i	\(0.816062\pi\)
\(31\)	5.70945	+	2.05330i	1.02545	+	0.368784i	0.796490	−	0.604652i	\(-0.206688\pi\)
							0.228958	+	0.973436i	\(0.426468\pi\)
\(37\)	−4.50691	+	3.63676i	−0.740931	+	0.597880i	−0.922726	−	0.385457i	\(-0.874044\pi\)
							0.181795	+	0.983337i	\(0.441809\pi\)
\(41\)	−3.66236	−	6.34339i	−0.571964	−	0.990671i	−0.996364	−	0.0851959i	\(-0.972848\pi\)
							0.424400	−	0.905475i	\(-0.360485\pi\)
\(43\)	−0.277303	+	2.82592i	−0.0422883	+	0.430949i	0.950470	+	0.310817i	\(0.100603\pi\)
							−0.992758	+	0.120132i	\(0.961668\pi\)
\(47\)	4.27485	−	0.345102i	0.623551	−	0.0503382i	0.235347	−	0.971912i	\(-0.424377\pi\)
							0.388204	+	0.921573i	\(0.373095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.15	✓	6480
547.137	even	273	inner	547.2.o.a.137.15	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.15	✓	6480	1.1	even	1	trivial
547.2.o.a.137.15	yes	6480	547.137	even	273	inner