Embedded newform 547.2.j.a.11.3

• Label: 547.2.j.a.11.3
• Level: $547$
• Weight: $2$
• Character: 547.11
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.j (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52033 - 1.07381i) q^{2} -3.03640 q^{3} +(3.81355 + 3.97033i) q^{4} +(-0.333001 + 0.0541179i) q^{5} +(7.65274 + 3.26053i) q^{6} +(-2.25223 - 1.69244i) q^{7} +(-3.40511 - 8.97854i) q^{8} +6.21972 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52033	−	1.07381i	−1.78214	−	0.759301i	−0.988558	−	0.150843i	\(-0.951801\pi\)
							−0.793586	−	0.608457i	\(-0.791789\pi\)
\(3\)	−3.03640			−1.75307			−0.876533	−	0.481341i	\(-0.840150\pi\)
							−0.876533	+	0.481341i	\(0.840150\pi\)
\(5\)	−0.333001	+	0.0541179i	−0.148922	+	0.0242023i	−0.234415	−	0.972137i	\(-0.575318\pi\)
							0.0854929	+	0.996339i	\(0.472753\pi\)
\(7\)	−2.25223	−	1.69244i	−0.851263	−	0.639683i	0.0828886	−	0.996559i	\(-0.473585\pi\)
							−0.934152	+	0.356876i	\(0.883842\pi\)
\(11\)	−5.22091	+	0.421475i	−1.57416	+	0.127080i	−0.836453	−	0.548038i	\(-0.815375\pi\)
							−0.737710	+	0.675118i	\(0.764093\pi\)
\(13\)	−2.38905	+	4.13795i	−0.662603	+	1.14766i	0.317327	+	0.948316i	\(0.397215\pi\)
							−0.979929	+	0.199345i	\(0.936118\pi\)
\(17\)	−0.901042	−	1.89891i	−0.218535	−	0.460552i	0.764343	−	0.644810i	\(-0.223063\pi\)
							−0.982878	+	0.184257i	\(0.941012\pi\)
\(19\)	0.00270668	+	0.0671656i	0.000620956	+	0.0154089i	0.999470	−	0.0325603i	\(-0.0103661\pi\)
							−0.998849	+	0.0479692i	\(0.984725\pi\)
\(23\)	−0.542104	+	2.65540i	−0.113036	+	0.553689i	0.883347	+	0.468719i	\(0.155284\pi\)
							−0.996384	+	0.0849697i	\(0.972921\pi\)
\(29\)	−1.94136	+	1.71990i	−0.360502	+	0.319377i	−0.823859	−	0.566795i	\(-0.808183\pi\)
							0.463357	+	0.886172i	\(0.346645\pi\)
\(31\)	−4.45750	+	2.33948i	−0.800590	+	0.420182i	−0.814818	−	0.579717i	\(-0.803163\pi\)
							0.0142279	+	0.999899i	\(0.495471\pi\)
\(37\)	1.68755	+	8.26615i	0.277431	+	1.35895i	0.844623	+	0.535361i	\(0.179824\pi\)
							−0.567193	+	0.823585i	\(0.691971\pi\)
\(41\)	4.21480	−	7.30025i	0.658242	−	1.14011i	−0.322829	−	0.946457i	\(-0.604634\pi\)
							0.981071	−	0.193651i	\(-0.0620329\pi\)
\(43\)	1.22396	−	4.22560i	0.186652	−	0.644399i	−0.811540	−	0.584297i	\(-0.801370\pi\)
							0.998192	−	0.0601016i	\(-0.0191425\pi\)
\(47\)	−7.22802	−	2.41307i	−1.05431	−	0.351982i	−0.263898	−	0.964551i	\(-0.585008\pi\)
							−0.790417	+	0.612569i	\(0.790136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.3	✓	1080
547.199	even	39	inner	547.2.j.a.199.3	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.3	✓	1080	1.1	even	1	trivial
547.2.j.a.199.3	yes	1080	547.199	even	39	inner