Embedded newform 27.6.e.a.4.2

• Label: 27.6.e.a.4.2
• Level: $27$
• Weight: $6$
• Character: 27.4
• Analytic conductor: $4.330$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.33036313495\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	27.4
Dual form			27.6.e.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.79489 - 3.20108i) q^{2} +(13.9939 - 6.86812i) q^{3} +(42.5898 + 35.7371i) q^{4} +(-11.0981 + 62.9404i) q^{5} +(-145.060 + 15.6088i) q^{6} +(158.627 - 133.104i) q^{7} +(-110.426 - 191.263i) q^{8} +(148.658 - 192.223i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.79489	−	3.20108i	−1.55473	−	0.565876i	−0.585210	−	0.810882i	\(-0.698988\pi\)
							−0.969522	+	0.245006i	\(0.921210\pi\)
\(3\)	13.9939	−	6.86812i	0.897708	−	0.440590i
							\(5\)	−11.0981	+	62.9404i	−0.198529	+	1.12591i	0.708774	+	0.705435i	\(0.249248\pi\)
−0.907303	+	0.420477i	\(0.861863\pi\)
\(7\)	158.627	−	133.104i	1.22358	−	1.02670i	0.224949	−	0.974371i	\(-0.427778\pi\)
							0.998630	−	0.0523340i	\(-0.0166660\pi\)
\(11\)	−38.4490	−	218.055i	−0.0958084	−	0.543356i	−0.994497	−	0.104768i	\(-0.966590\pi\)
							0.898688	−	0.438588i	\(-0.144521\pi\)
\(13\)	10.0542	−	3.65943i	0.0165002	−	0.00600559i	−0.333757	−	0.942659i	\(-0.608316\pi\)
							0.350257	+	0.936654i	\(0.386094\pi\)
\(17\)	869.206	−	1505.51i	0.729458	−	1.26346i	−0.227654	−	0.973742i	\(-0.573106\pi\)
							0.957112	−	0.289717i	\(-0.0935611\pi\)
\(19\)	1055.46	+	1828.10i	0.670743	+	1.16176i	0.977694	+	0.210036i	\(0.0673580\pi\)
							−0.306951	+	0.951725i	\(0.599309\pi\)
\(23\)	2064.10	+	1731.98i	0.813598	+	0.682690i	0.951464	−	0.307761i	\(-0.0995797\pi\)
							−0.137865	+	0.990451i	\(0.544024\pi\)
\(29\)	−2509.43	−	913.356i	−0.554089	−	0.201672i	0.0497735	−	0.998761i	\(-0.484150\pi\)
							−0.603862	+	0.797089i	\(0.706372\pi\)
\(31\)	−4665.50	−	3914.82i	−0.871955	−	0.731657i	0.0925539	−	0.995708i	\(-0.470497\pi\)
							−0.964509	+	0.264051i	\(0.914941\pi\)
\(37\)	−2834.39	+	4909.31i	−0.340373	+	0.589544i	−0.984502	−	0.175373i	\(-0.943887\pi\)
							0.644129	+	0.764917i	\(0.277220\pi\)
\(41\)	7539.51	−	2744.16i	0.700460	−	0.254947i	0.0328526	−	0.999460i	\(-0.489541\pi\)
							0.667607	+	0.744514i	\(0.267319\pi\)
\(43\)	1914.18	+	10855.8i	0.157874	+	0.895348i	0.956111	+	0.293005i	\(0.0946551\pi\)
							−0.798237	+	0.602343i	\(0.794234\pi\)
\(47\)	−9340.08	+	7837.26i	−0.616746	+	0.517511i	−0.896779	−	0.442479i	\(-0.854099\pi\)
							0.280033	+	0.959990i	\(0.409655\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.4.2	✓	84
3.2	odd	2		81.6.e.a.64.13		84
27.7	even	9	inner	27.6.e.a.7.2	yes	84
27.13	even	9		729.6.a.c.1.39		42
27.14	odd	18		729.6.a.e.1.4		42
27.20	odd	18		81.6.e.a.19.13		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.2	✓	84	1.1	even	1	trivial
27.6.e.a.7.2	yes	84	27.7	even	9	inner
81.6.e.a.19.13		84	27.20	odd	18
81.6.e.a.64.13		84	3.2	odd	2
729.6.a.c.1.39		42	27.13	even	9
729.6.a.e.1.4		42	27.14	odd	18