Embedded newform 547.2.j.a.11.12

• Label: 547.2.j.a.11.12
• 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			11.12
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41644 - 0.603488i) q^{2} +0.684807 q^{3} +(0.256650 + 0.267201i) q^{4} +(-2.74804 + 0.446599i) q^{5} +(-0.969987 - 0.413273i) q^{6} +(3.82197 + 2.87203i) q^{7} +(0.889652 + 2.34582i) q^{8} -2.53104 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41644	−	0.603488i	−1.00157	−	0.426730i	−0.171981	−	0.985100i	\(-0.555017\pi\)
							−0.829592	+	0.558370i	\(0.811427\pi\)
\(3\)	0.684807			0.395374			0.197687	−	0.980265i	\(-0.436657\pi\)
							0.197687	+	0.980265i	\(0.436657\pi\)
\(5\)	−2.74804	+	0.446599i	−1.22896	+	0.199725i	−0.740015	−	0.672590i	\(-0.765182\pi\)
							−0.488944	+	0.872315i	\(0.662618\pi\)
\(7\)	3.82197	+	2.87203i	1.44457	+	1.08552i	0.979774	+	0.200105i	\(0.0641284\pi\)
							0.464795	+	0.885418i	\(0.346128\pi\)
\(11\)	−2.92512	+	0.236140i	−0.881958	+	0.0711990i	−0.513142	−	0.858304i	\(-0.671519\pi\)
							−0.368816	+	0.929503i	\(0.620237\pi\)
\(13\)	1.90219	−	3.29469i	0.527572	−	0.913782i	−0.471911	−	0.881646i	\(-0.656436\pi\)
							0.999484	−	0.0321360i	\(-0.0102310\pi\)
\(17\)	−2.36530	−	4.98476i	−0.573669	−	1.20898i	−0.958261	−	0.285894i	\(-0.907709\pi\)
							0.384593	−	0.923086i	\(-0.374342\pi\)
\(19\)	−0.198031	−	4.91408i	−0.0454313	−	1.12737i	−0.847725	−	0.530436i	\(-0.822028\pi\)
							0.802293	−	0.596930i	\(-0.203613\pi\)
\(23\)	0.698858	−	3.42323i	0.145722	−	0.713794i	−0.839616	−	0.543180i	\(-0.817220\pi\)
							0.985338	−	0.170613i	\(-0.0545749\pi\)
\(29\)	−5.59951	+	4.96074i	−1.03980	+	0.921186i	−0.996961	−	0.0778961i	\(-0.975180\pi\)
							−0.0428423	+	0.999082i	\(0.513641\pi\)
\(31\)	−8.46775	+	4.44422i	−1.52085	+	0.798205i	−0.998512	−	0.0545315i	\(-0.982633\pi\)
							−0.522341	+	0.852737i	\(0.674941\pi\)
\(37\)	0.190331	+	0.932300i	0.0312901	+	0.153269i	0.992434	−	0.122778i	\(-0.0391804\pi\)
							−0.961144	+	0.276048i	\(0.910975\pi\)
\(41\)	0.529896	−	0.917807i	0.0827558	−	0.143337i	−0.821677	−	0.569954i	\(-0.806961\pi\)
							0.904433	+	0.426616i	\(0.140295\pi\)
\(43\)	2.07592	−	7.16692i	0.316575	−	1.09294i	−0.631626	−	0.775274i	\(-0.717612\pi\)
							0.948201	−	0.317671i	\(-0.102901\pi\)
\(47\)	−0.614502	−	0.205151i	−0.0896344	−	0.0299243i	0.271497	−	0.962439i	\(-0.412481\pi\)
							−0.361132	+	0.932515i	\(0.617609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.12	✓	1080
547.199	even	39	inner	547.2.j.a.199.12	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.12	✓	1080	1.1	even	1	trivial
547.2.j.a.199.12	yes	1080	547.199	even	39	inner