Embedded newform 547.2.o.a.4.11

• Label: 547.2.o.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82406 - 0.0209916i) q^{2} +(-0.585104 + 2.56351i) q^{3} +(1.32728 + 0.0305533i) q^{4} +(-1.34163 + 2.52090i) q^{5} +(1.12108 - 4.66371i) q^{6} +(-0.483254 - 1.19083i) q^{7} +(1.22578 + 0.0423345i) q^{8} +(-3.52632 - 1.69819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82406	−	0.0209916i	−1.28981	−	0.0148433i	−0.636108	−	0.771600i	\(-0.719457\pi\)
							−0.653697	+	0.756756i	\(0.726783\pi\)
\(3\)	−0.585104	+	2.56351i	−0.337810	+	1.48004i	0.465802	+	0.884889i	\(0.345766\pi\)
							−0.803612	+	0.595153i	\(0.797091\pi\)
\(5\)	−1.34163	+	2.52090i	−0.599993	+	1.12738i	0.379365	+	0.925247i	\(0.376142\pi\)
							−0.979358	+	0.202132i	\(0.935213\pi\)
\(7\)	−0.483254	−	1.19083i	−0.182653	−	0.450092i	0.807538	−	0.589816i	\(-0.200800\pi\)
							−0.990190	+	0.139724i	\(0.955378\pi\)
\(11\)	−4.19869	−	4.37130i	−1.26595	−	1.31800i	−0.925985	−	0.377561i	\(-0.876763\pi\)
							−0.339969	−	0.940437i	\(-0.610417\pi\)
\(13\)	4.43866	+	0.669020i	1.23106	+	0.185553i	0.732176	−	0.681115i	\(-0.238505\pi\)
							0.498885	+	0.866668i	\(0.333743\pi\)
\(17\)	1.03622	−	6.15233i	0.251320	−	1.49216i	−0.519151	−	0.854683i	\(-0.673752\pi\)
							0.770471	−	0.637475i	\(-0.220021\pi\)
\(19\)	−0.803415	+	0.433529i	−0.184316	+	0.0994584i	−0.564429	−	0.825482i	\(-0.690904\pi\)
							0.380113	+	0.924940i	\(0.375885\pi\)
\(23\)	−3.89912	−	0.913584i	−0.813023	−	0.190496i	−0.199220	−	0.979955i	\(-0.563841\pi\)
							−0.613803	+	0.789459i	\(0.710361\pi\)
\(29\)	−2.04301	−	1.16391i	−0.379377	−	0.216132i	0.293380	−	0.955996i	\(-0.405220\pi\)
							−0.672757	+	0.739863i	\(0.734890\pi\)
\(31\)	6.95289	+	2.50048i	1.24878	+	0.449100i	0.877569	−	0.479450i	\(-0.159164\pi\)
							0.371207	+	0.928550i	\(0.378944\pi\)
\(37\)	−3.02909	+	2.44426i	−0.497979	+	0.401834i	−0.843959	−	0.536408i	\(-0.819781\pi\)
							0.345980	+	0.938242i	\(0.387546\pi\)
\(41\)	−2.40457	−	4.16484i	−0.375531	−	0.650438i	0.614876	−	0.788624i	\(-0.289206\pi\)
							−0.990406	+	0.138186i	\(0.955873\pi\)
\(43\)	0.0164151	−	0.167282i	0.00250328	−	0.0255102i	−0.993887	−	0.110406i	\(-0.964785\pi\)
							0.996390	+	0.0848961i	\(0.0270558\pi\)
\(47\)	−0.355226	+	0.0286768i	−0.0518150	+	0.00418294i	−0.106347	−	0.994329i	\(-0.533915\pi\)
							0.0545319	+	0.998512i	\(0.482633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.11	✓	6480
547.137	even	273	inner	547.2.o.a.137.11	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.11	✓	6480	1.1	even	1	trivial
547.2.o.a.137.11	yes	6480	547.137	even	273	inner