Embedded newform 547.2.o.a.4.8

• Label: 547.2.o.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.07856 - 0.0239204i) q^{2} +(0.521570 - 2.28515i) q^{3} +(2.32035 + 0.0534131i) q^{4} +(-0.897784 + 1.68692i) q^{5} +(-1.13877 + 4.73733i) q^{6} +(-0.603206 - 1.48642i) q^{7} +(-0.666796 - 0.0230290i) q^{8} +(-2.24696 - 1.08208i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.07856	−	0.0239204i	−1.46976	−	0.0169143i	−0.727077	−	0.686556i	\(-0.759122\pi\)
							−0.742684	+	0.669642i	\(0.766448\pi\)
\(3\)	0.521570	−	2.28515i	0.301129	−	1.31933i	−0.567298	−	0.823513i	\(-0.692011\pi\)
							0.868426	−	0.495818i	\(-0.165132\pi\)
\(5\)	−0.897784	+	1.68692i	−0.401501	+	0.754416i	−0.998869	−	0.0475519i	\(-0.984858\pi\)
							0.597368	+	0.801968i	\(0.296213\pi\)
\(7\)	−0.603206	−	1.48642i	−0.227990	−	0.561812i	0.769008	−	0.639239i	\(-0.220751\pi\)
							−0.996998	+	0.0774271i	\(0.975329\pi\)
\(11\)	2.08454	+	2.17024i	0.628513	+	0.654352i	0.956977	−	0.290165i	\(-0.0937102\pi\)
							−0.328463	+	0.944517i	\(0.606531\pi\)
\(13\)	−4.56605	−	0.688221i	−1.26639	−	0.190878i	−0.518720	−	0.854944i	\(-0.673591\pi\)
							−0.747674	+	0.664066i	\(0.768829\pi\)
\(17\)	0.862167	−	5.11893i	0.209106	−	1.24152i	−0.661652	−	0.749811i	\(-0.730144\pi\)
							0.870758	−	0.491711i	\(-0.163628\pi\)
\(19\)	−5.01270	+	2.70489i	−1.14999	+	0.620544i	−0.934501	−	0.355961i	\(-0.884153\pi\)
							−0.215491	+	0.976506i	\(0.569135\pi\)
\(23\)	−6.98572	−	1.63679i	−1.45662	−	0.341294i	−0.576914	−	0.816805i	\(-0.695743\pi\)
							−0.879710	+	0.475511i	\(0.842263\pi\)
\(29\)	−1.25427	−	0.714562i	−0.232912	−	0.132691i	0.374087	−	0.927394i	\(-0.377956\pi\)
							−0.606999	+	0.794703i	\(0.707627\pi\)
\(31\)	−7.08987	−	2.54975i	−1.27338	−	0.457948i	−0.387502	−	0.921869i	\(-0.626662\pi\)
							−0.885877	+	0.463921i	\(0.846442\pi\)
\(37\)	−3.13933	+	2.53322i	−0.516102	+	0.416458i	−0.850501	−	0.525973i	\(-0.823701\pi\)
							0.334399	+	0.942432i	\(0.391467\pi\)
\(41\)	4.23812	+	7.34064i	0.661884	+	1.14642i	0.980120	+	0.198405i	\(0.0635760\pi\)
							−0.318237	+	0.948011i	\(0.603091\pi\)
\(43\)	−0.873343	+	8.90001i	−0.133184	+	1.35724i	0.661377	+	0.750054i	\(0.269972\pi\)
							−0.794560	+	0.607185i	\(0.792299\pi\)
\(47\)	1.05829	−	0.0854340i	0.154367	−	0.0124618i	−0.00304122	−	0.999995i	\(-0.500968\pi\)
							0.157409	+	0.987534i	\(0.449686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.8	✓	6480
547.137	even	273	inner	547.2.o.a.137.8	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.8	✓	6480	1.1	even	1	trivial
547.2.o.a.137.8	yes	6480	547.137	even	273	inner