Embedded newform 27.6.e.a.7.10

• Label: 27.6.e.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	27.7
Dual form			27.6.e.a.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.64663 - 1.69124i) q^{2} +(-11.1566 + 10.8872i) q^{3} +(-5.78253 + 4.85212i) q^{4} +(9.66519 + 54.8140i) q^{5} +(-33.4278 + 69.4570i) q^{6} +(60.6928 + 50.9273i) q^{7} +(-97.7806 + 169.361i) q^{8} +(5.93943 - 242.927i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.64663	−	1.69124i	0.821416	−	0.298971i	0.103085	−	0.994673i	\(-0.467129\pi\)
							0.718331	+	0.695702i	\(0.244906\pi\)
\(3\)	−11.1566	+	10.8872i	−0.715696	+	0.698412i
							\(5\)	9.66519	+	54.8140i	0.172896	+	0.980543i	0.940544	+	0.339671i	\(0.110316\pi\)
−0.767648	+	0.640872i	\(0.778573\pi\)
\(7\)	60.6928	+	50.9273i	0.468158	+	0.392831i	0.846122	−	0.532989i	\(-0.178931\pi\)
							−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	30.3601	−	172.181i	0.0756522	−	0.429045i	−0.923333	−	0.384001i	\(-0.874546\pi\)
							0.998985	−	0.0450441i	\(-0.0143428\pi\)
\(13\)	162.354	+	59.0920i	0.266443	+	0.0969774i	0.471787	−	0.881712i	\(-0.343609\pi\)
							−0.205344	+	0.978690i	\(0.565831\pi\)
\(17\)	263.395	+	456.213i	0.221047	+	0.382865i	0.955126	−	0.296199i	\(-0.0957193\pi\)
							−0.734079	+	0.679064i	\(0.762386\pi\)
\(19\)	−343.270	+	594.561i	−0.218148	+	0.377844i	−0.954242	−	0.299036i	\(-0.903335\pi\)
							0.736094	+	0.676880i	\(0.236668\pi\)
\(23\)	3501.96	−	2938.49i	1.38036	−	1.15826i	0.411278	−	0.911510i	\(-0.365082\pi\)
							0.969080	−	0.246748i	\(-0.0793620\pi\)
\(29\)	4147.67	−	1509.63i	0.915818	−	0.333330i	0.159245	−	0.987239i	\(-0.449094\pi\)
							0.756573	+	0.653909i	\(0.226872\pi\)
\(31\)	−8175.51	+	6860.06i	−1.52795	+	1.28211i	−0.716787	+	0.697292i	\(0.754388\pi\)
							−0.811168	+	0.584814i	\(0.801168\pi\)
\(37\)	7714.90	+	13362.6i	0.926459	+	1.60467i	0.789198	+	0.614139i	\(0.210497\pi\)
							0.137261	+	0.990535i	\(0.456170\pi\)
\(41\)	5165.96	+	1880.26i	0.479945	+	0.174686i	0.570652	−	0.821192i	\(-0.306690\pi\)
							−0.0907070	+	0.995878i	\(0.528913\pi\)
\(43\)	1242.19	−	7044.79i	0.102451	−	0.581028i	−0.889757	−	0.456435i	\(-0.849126\pi\)
							0.992208	−	0.124593i	\(-0.0397627\pi\)
\(47\)	−1340.96	−	1125.20i	−0.0885462	−	0.0742991i	0.597440	−	0.801914i	\(-0.296185\pi\)
							−0.685986	+	0.727614i	\(0.740629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.7.10	yes	84
3.2	odd	2		81.6.e.a.19.5		84
27.2	odd	18		729.6.a.e.1.29		42
27.4	even	9	inner	27.6.e.a.4.10	✓	84
27.23	odd	18		81.6.e.a.64.5		84
27.25	even	9		729.6.a.c.1.14		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.10	✓	84	27.4	even	9	inner
27.6.e.a.7.10	yes	84	1.1	even	1	trivial
81.6.e.a.19.5		84	3.2	odd	2
81.6.e.a.64.5		84	27.23	odd	18
729.6.a.c.1.14		42	27.25	even	9
729.6.a.e.1.29		42	27.2	odd	18