Embedded newform 547.2.j.a.11.15

• Label: 547.2.j.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.985454 - 0.419863i) q^{2} -2.18181 q^{3} +(-0.590614 - 0.614895i) q^{4} +(-1.42571 + 0.231701i) q^{5} +(2.15008 + 0.916062i) q^{6} +(0.947050 + 0.711662i) q^{7} +(1.08354 + 2.85705i) q^{8} +1.76031 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.985454	−	0.419863i	−0.696821	−	0.296888i	0.0143597	−	0.999897i	\(-0.495429\pi\)
							−0.711181	+	0.703009i	\(0.751839\pi\)
\(3\)	−2.18181			−1.25967			−0.629836	−	0.776729i	\(-0.716878\pi\)
							−0.629836	+	0.776729i	\(0.716878\pi\)
\(5\)	−1.42571	+	0.231701i	−0.637597	+	0.103620i	−0.470613	−	0.882340i	\(-0.655967\pi\)
							−0.166985	+	0.985960i	\(0.553403\pi\)
\(7\)	0.947050	+	0.711662i	0.357951	+	0.268983i	0.764470	−	0.644660i	\(-0.223001\pi\)
							−0.406519	+	0.913643i	\(0.633257\pi\)
\(11\)	−2.13546	+	0.172392i	−0.643867	+	0.0519783i	−0.398087	−	0.917348i	\(-0.630326\pi\)
							−0.245779	+	0.969326i	\(0.579044\pi\)
\(13\)	−1.22646	+	2.12430i	−0.340160	+	0.589174i	−0.984462	−	0.175597i	\(-0.943814\pi\)
							0.644302	+	0.764771i	\(0.277148\pi\)
\(17\)	0.477148	+	1.00557i	0.115725	+	0.243886i	0.953046	−	0.302827i	\(-0.0979303\pi\)
							−0.837320	+	0.546713i	\(0.815879\pi\)
\(19\)	0.213512	+	5.29825i	0.0489830	+	1.21550i	0.817598	+	0.575790i	\(0.195305\pi\)
							−0.768615	+	0.639712i	\(0.779054\pi\)
\(23\)	1.77518	−	8.69538i	0.370150	−	1.81311i	−0.186604	−	0.982435i	\(-0.559748\pi\)
							0.556754	−	0.830677i	\(-0.312047\pi\)
\(29\)	6.19410	−	5.48749i	1.15022	−	1.01900i	0.150678	−	0.988583i	\(-0.451854\pi\)
							0.999537	−	0.0304190i	\(-0.00968416\pi\)
\(31\)	−6.95083	+	3.64808i	−1.24841	+	0.655214i	−0.953811	−	0.300408i	\(-0.902877\pi\)
							−0.294596	+	0.955622i	\(0.595185\pi\)
\(37\)	−0.663498	−	3.25003i	−0.109078	−	0.534301i	−0.997206	−	0.0747037i	\(-0.976199\pi\)
							0.888127	−	0.459597i	\(-0.152006\pi\)
\(41\)	2.02659	−	3.51016i	0.316500	−	0.548194i	−0.663255	−	0.748393i	\(-0.730826\pi\)
							0.979755	+	0.200199i	\(0.0641589\pi\)
\(43\)	3.19180	−	11.0194i	0.486745	−	1.68044i	−0.222059	−	0.975033i	\(-0.571278\pi\)
							0.708804	−	0.705405i	\(-0.249235\pi\)
\(47\)	−5.49747	−	1.83532i	−0.801888	−	0.267709i	−0.113957	−	0.993486i	\(-0.536353\pi\)
							−0.687931	+	0.725776i	\(0.741481\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.15	✓	1080
547.199	even	39	inner	547.2.j.a.199.15	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.15	✓	1080	1.1	even	1	trivial
547.2.j.a.199.15	yes	1080	547.199	even	39	inner