Embedded newform 27.6.e.a.7.2

• Label: 27.6.e.a.7.2
• Level: $27$
• Weight: $6$
• Character: 27.7
• 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			7.2
Character	\(\chi\)	\(=\)	27.7
Dual form			27.6.e.a.4.2

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.2	✓	84	27.4	even	9	inner
27.6.e.a.7.2	yes	84	1.1	even	1	trivial
81.6.e.a.19.13		84	3.2	odd	2
81.6.e.a.64.13		84	27.23	odd	18
729.6.a.c.1.39		42	27.25	even	9
729.6.a.e.1.4		42	27.2	odd	18