Embedded newform 547.2.j.a.11.17

• Label: 547.2.j.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.877457 - 0.373850i) q^{2} +3.24938 q^{3} +(-0.755281 - 0.786331i) q^{4} +(-3.27067 + 0.531536i) q^{5} +(-2.85119 - 1.21478i) q^{6} +(1.56237 + 1.17405i) q^{7} +(1.04519 + 2.75593i) q^{8} +7.55848 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.877457	−	0.373850i	−0.620456	−	0.264352i	0.0587807	−	0.998271i	\(-0.481279\pi\)
							−0.679237	+	0.733919i	\(0.737689\pi\)
\(3\)	3.24938			1.87603			0.938016	−	0.346593i	\(-0.112662\pi\)
							0.938016	+	0.346593i	\(0.112662\pi\)
\(5\)	−3.27067	+	0.531536i	−1.46269	+	0.237710i	−0.839072	−	0.544020i	\(-0.816902\pi\)
							−0.623617	+	0.781730i	\(0.714337\pi\)
\(7\)	1.56237	+	1.17405i	0.590522	+	0.443748i	0.853529	−	0.521046i	\(-0.174458\pi\)
							−0.263007	+	0.964794i	\(0.584714\pi\)
\(11\)	5.88826	−	0.475349i	1.77538	−	0.143323i	0.851090	−	0.525021i	\(-0.175942\pi\)
							0.924287	+	0.381697i	\(0.124660\pi\)
\(13\)	−2.83071	+	4.90293i	−0.785097	+	1.35983i	0.143844	+	0.989600i	\(0.454054\pi\)
							−0.928941	+	0.370227i	\(0.879280\pi\)
\(17\)	0.511582	+	1.07814i	0.124077	+	0.261486i	0.955979	−	0.293437i	\(-0.0947990\pi\)
							−0.831902	+	0.554923i	\(0.812748\pi\)
\(19\)	−0.123289	−	3.05938i	−0.0282844	−	0.701870i	−0.949676	−	0.313234i	\(-0.898587\pi\)
							0.921391	−	0.388636i	\(-0.127054\pi\)
\(23\)	0.824216	−	4.03728i	0.171861	−	0.841830i	−0.798841	−	0.601542i	\(-0.794553\pi\)
							0.970702	−	0.240288i	\(-0.0772418\pi\)
\(29\)	4.36752	−	3.86929i	0.811028	−	0.718508i	−0.151862	−	0.988402i	\(-0.548527\pi\)
							0.962890	+	0.269893i	\(0.0869884\pi\)
\(31\)	−1.98039	+	1.03939i	−0.355688	+	0.186680i	−0.633097	−	0.774072i	\(-0.718217\pi\)
							0.277409	+	0.960752i	\(0.410524\pi\)
\(37\)	−0.0862842	−	0.422648i	−0.0141850	−	0.0694829i	0.972082	−	0.234641i	\(-0.0753916\pi\)
							−0.986267	+	0.165158i	\(0.947186\pi\)
\(41\)	−1.76755	+	3.06149i	−0.276046	+	0.478125i	−0.970398	−	0.241510i	\(-0.922357\pi\)
							0.694353	+	0.719635i	\(0.255691\pi\)
\(43\)	0.231041	−	0.797648i	0.0352335	−	0.121640i	−0.940974	−	0.338479i	\(-0.890088\pi\)
							0.976207	+	0.216839i	\(0.0695747\pi\)
\(47\)	6.21775	+	2.07579i	0.906951	+	0.302785i	0.731610	−	0.681723i	\(-0.238769\pi\)
							0.175341	+	0.984508i	\(0.443897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.17	✓	1080
547.199	even	39	inner	547.2.j.a.199.17	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.17	✓	1080	1.1	even	1	trivial
547.2.j.a.199.17	yes	1080	547.199	even	39	inner