Embedded newform 108.5.j.a.7.13

• Label: 108.5.j.a.7.13
• Level: $108$
• Weight: $5$
• Character: 108.7
• Analytic conductor: $11.164$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1639560131\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			7.13
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.43437 - 2.05063i) q^{2} +(0.424142 - 8.99000i) q^{3} +(7.58985 + 14.0852i) q^{4} +(2.37776 + 13.4850i) q^{5} +(-19.8918 + 30.0053i) q^{6} +(18.1869 - 21.6743i) q^{7} +(2.81722 - 63.9380i) q^{8} +(-80.6402 - 7.62607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.43437	−	2.05063i	−0.858594	−	0.512657i
							\(3\)	0.424142	−	8.99000i	0.0471269	−	0.998889i
\(5\)	2.37776	+	13.4850i	0.0951106	+	0.539399i								0.994713	+	0.102690i	\(0.0327449\pi\)
							−0.899603	+	0.436709i	\(0.856144\pi\)
\(7\)	18.1869	−	21.6743i	0.371162	−	0.442333i	−0.547842	−	0.836582i	\(-0.684551\pi\)
							0.919004	+	0.394248i	\(0.128995\pi\)
\(11\)	196.406	+	34.6317i	1.62319	+	0.286212i	0.909953	−	0.414711i	\(-0.136117\pi\)
							0.713237	+	0.700923i	\(0.247228\pi\)
\(13\)	216.047	+	78.6348i	1.27839	+	0.465295i	0.889898	−	0.456159i	\(-0.150775\pi\)
							0.388489	+	0.921453i	\(0.372997\pi\)
\(17\)	−243.120	−	421.095i	−0.841244	−	1.45708i	−0.888844	−	0.458211i	\(-0.848490\pi\)
							0.0475996	−	0.998866i	\(-0.484843\pi\)
\(19\)	157.194	+	90.7557i	0.435439	+	0.251401i	0.701661	−	0.712511i	\(-0.252442\pi\)
							−0.266222	+	0.963912i	\(0.585775\pi\)
\(23\)	−603.789	−	719.568i	−1.14138	−	1.36024i	−0.923194	−	0.384335i	\(-0.874431\pi\)
							−0.218185	−	0.975907i	\(-0.570014\pi\)
\(29\)	60.8337	−	22.1417i	0.0723350	−	0.0263278i	−0.305599	−	0.952160i	\(-0.598857\pi\)
							0.377934	+	0.925832i	\(0.376635\pi\)
\(31\)	−40.4212	−	48.1721i	−0.0420616	−	0.0501271i	0.744604	−	0.667507i	\(-0.232639\pi\)
							−0.786665	+	0.617380i	\(0.788194\pi\)
\(37\)	−907.999	−	1572.70i	−0.663257	−	1.14879i	−0.979755	−	0.200202i	\(-0.935840\pi\)
							0.316498	−	0.948593i	\(-0.397493\pi\)
\(41\)	−1806.02	−	657.336i	−1.07437	−	0.391039i	−0.256561	−	0.966528i	\(-0.582589\pi\)
							−0.817809	+	0.575489i	\(0.804812\pi\)
\(43\)	1619.21	+	285.510i	0.875722	+	0.154413i	0.593401	−	0.804907i	\(-0.297785\pi\)
							0.282321	+	0.959320i	\(0.408896\pi\)
\(47\)	1082.88	−	1290.53i	0.490213	−	0.584213i	−0.463059	−	0.886328i	\(-0.653248\pi\)
							0.953272	+	0.302115i	\(0.0976925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.13	✓	420
4.3	odd	2	inner	108.5.j.a.7.28	yes	420
27.4	even	9	inner	108.5.j.a.31.28	yes	420
108.31	odd	18	inner	108.5.j.a.31.13	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.13	✓	420	1.1	even	1	trivial
108.5.j.a.7.28	yes	420	4.3	odd	2	inner
108.5.j.a.31.13	yes	420	108.31	odd	18	inner
108.5.j.a.31.28	yes	420	27.4	even	9	inner