Embedded newform 547.2.j.a.11.5

• Label: 547.2.j.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.12444 - 0.905139i) q^{2} -0.527263 q^{3} +(2.30852 + 2.40342i) q^{4} +(-3.39503 + 0.551746i) q^{5} +(1.12014 + 0.477247i) q^{6} +(0.0185396 + 0.0139316i) q^{7} +(-1.09115 - 2.87714i) q^{8} -2.72199 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.12444	−	0.905139i	−1.50221	−	0.640030i	−0.524767	−	0.851246i	\(-0.675847\pi\)
							−0.977439	+	0.211216i	\(0.932258\pi\)
\(3\)	−0.527263			−0.304416			−0.152208	−	0.988349i	\(-0.548638\pi\)
							−0.152208	+	0.988349i	\(0.548638\pi\)
\(5\)	−3.39503	+	0.551746i	−1.51830	+	0.246748i	−0.861672	−	0.507465i	\(-0.830583\pi\)
							−0.656630	+	0.754213i	\(0.728019\pi\)
\(7\)	0.0185396	+	0.0139316i	0.00700730	+	0.00526564i	0.604240	−	0.796802i	\(-0.293477\pi\)
							−0.597233	+	0.802068i	\(0.703733\pi\)
\(11\)	2.01945	−	0.163027i	0.608888	−	0.0491545i	0.227823	−	0.973703i	\(-0.426839\pi\)
							0.381065	+	0.924548i	\(0.375557\pi\)
\(13\)	−2.91517	+	5.04922i	−0.808523	+	1.40040i	0.105364	+	0.994434i	\(0.466399\pi\)
							−0.913887	+	0.405969i	\(0.866934\pi\)
\(17\)	−1.81215	−	3.81903i	−0.439511	−	0.926250i	−0.995473	−	0.0950425i	\(-0.969701\pi\)
							0.555962	−	0.831208i	\(-0.312350\pi\)
\(19\)	0.285577	+	7.08651i	0.0655158	+	1.62576i	0.613797	+	0.789464i	\(0.289641\pi\)
							−0.548281	+	0.836294i	\(0.684718\pi\)
\(23\)	1.11336	−	5.45360i	0.232152	−	1.13715i	−0.681865	−	0.731478i	\(-0.738831\pi\)
							0.914017	−	0.405676i	\(-0.132964\pi\)
\(29\)	−3.73712	+	3.31080i	−0.693965	+	0.614800i	−0.934551	−	0.355828i	\(-0.884199\pi\)
							0.240586	+	0.970628i	\(0.422660\pi\)
\(31\)	8.29386	−	4.35295i	1.48962	−	0.781814i	0.493495	−	0.869748i	\(-0.335719\pi\)
							0.996126	+	0.0879347i	\(0.0280267\pi\)
\(37\)	−1.59687	−	7.82197i	−0.262523	−	1.28592i	−0.870579	−	0.492029i	\(-0.836255\pi\)
							0.608056	−	0.793894i	\(-0.291950\pi\)
\(41\)	2.82655	−	4.89572i	0.441432	−	0.764583i	−0.556364	−	0.830939i	\(-0.687804\pi\)
							0.997796	+	0.0663558i	\(0.0211372\pi\)
\(43\)	−2.29068	+	7.90835i	−0.349326	+	1.20601i	0.572986	+	0.819565i	\(0.305785\pi\)
							−0.922311	+	0.386447i	\(0.873702\pi\)
\(47\)	6.90180	+	2.30416i	1.00673	+	0.336096i	0.771765	−	0.635907i	\(-0.219374\pi\)
							0.234966	+	0.972004i	\(0.424502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.5	✓	1080
547.199	even	39	inner	547.2.j.a.199.5	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.5	✓	1080	1.1	even	1	trivial
547.2.j.a.199.5	yes	1080	547.199	even	39	inner