Embedded newform 108.5.j.a.7.17

• Label: 108.5.j.a.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.89182 - 2.76358i) q^{2} +(8.96055 - 0.841750i) q^{3} +(0.725269 + 15.9836i) q^{4} +(-0.256732 - 1.45600i) q^{5} +(-28.2386 - 22.3290i) q^{6} +(-22.3218 + 26.6021i) q^{7} +(42.0745 - 48.2259i) q^{8} +(79.5829 - 15.0851i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.89182	−	2.76358i	−0.722955	−	0.690895i
							\(3\)	8.96055	−	0.841750i	0.995617	−	0.0935278i
\(5\)	−0.256732	−	1.45600i	−0.0102693	−	0.0582400i								0.979242	−	0.202692i	\(-0.0649691\pi\)
							−0.989512	+	0.144452i	\(0.953858\pi\)
\(7\)	−22.3218	+	26.6021i	−0.455548	+	0.542901i	−0.944111	−	0.329628i	\(-0.893077\pi\)
							0.488563	+	0.872529i	\(0.337521\pi\)
\(11\)	92.1676	+	16.2516i	0.761716	+	0.134311i	0.540995	−	0.841026i	\(-0.318048\pi\)
							0.220721	+	0.975337i	\(0.429159\pi\)
\(13\)	−2.01935	−	0.734984i	−0.0119488	−	0.00434902i	0.336039	−	0.941848i	\(-0.390913\pi\)
							−0.347988	+	0.937499i	\(0.613135\pi\)
\(17\)	265.451	+	459.774i	0.918514	+	1.59091i	0.801673	+	0.597762i	\(0.203943\pi\)
							0.116841	+	0.993151i	\(0.462723\pi\)
\(19\)	198.676	+	114.706i	0.550349	+	0.317744i	0.749263	−	0.662273i	\(-0.230408\pi\)
							−0.198914	+	0.980017i	\(0.563741\pi\)
\(23\)	−172.973	−	206.141i	−0.326981	−	0.389681i	0.577361	−	0.816489i	\(-0.304083\pi\)
							−0.904342	+	0.426808i	\(0.859638\pi\)
\(29\)	1230.71	−	447.941i	1.46339	−	0.532629i	0.517089	−	0.855931i	\(-0.327015\pi\)
							0.946296	+	0.323303i	\(0.104793\pi\)
\(31\)	−247.550	−	295.018i	−0.257596	−	0.306991i	0.621711	−	0.783247i	\(-0.286438\pi\)
							−0.879307	+	0.476256i	\(0.841994\pi\)
\(37\)	−461.101	−	798.650i	−0.336816	−	0.583382i	0.647016	−	0.762476i	\(-0.276017\pi\)
							−0.983832	+	0.179094i	\(0.942683\pi\)
\(41\)	623.905	+	227.083i	0.371151	+	0.135088i	0.520860	−	0.853642i	\(-0.325611\pi\)
							−0.149709	+	0.988730i	\(0.547834\pi\)
\(43\)	−1420.29	−	250.436i	−0.768140	−	0.135444i	−0.224172	−	0.974550i	\(-0.571968\pi\)
							−0.543968	+	0.839106i	\(0.683079\pi\)
\(47\)	−1276.25	+	1520.97i	−0.577749	+	0.688534i	−0.973202	−	0.229952i	\(-0.926143\pi\)
							0.395453	+	0.918486i	\(0.370588\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.17	✓	420
4.3	odd	2	inner	108.5.j.a.7.33	yes	420
27.4	even	9	inner	108.5.j.a.31.33	yes	420
108.31	odd	18	inner	108.5.j.a.31.17	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.17	✓	420	1.1	even	1	trivial
108.5.j.a.7.33	yes	420	4.3	odd	2	inner
108.5.j.a.31.17	yes	420	108.31	odd	18	inner
108.5.j.a.31.33	yes	420	27.4	even	9	inner