Embedded newform 27.6.e.a.4.5

• Label: 27.6.e.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	27.4
Dual form			27.6.e.a.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.77365 - 1.73746i) q^{2} +(11.5896 + 10.4250i) q^{3} +(-4.74451 - 3.98112i) q^{4} +(-2.77896 + 15.7603i) q^{5} +(-37.2117 - 69.9018i) q^{6} +(-95.3445 + 80.0035i) q^{7} +(97.0117 + 168.029i) q^{8} +(25.6389 + 241.644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.77365	−	1.73746i	−0.843869	−	0.307143i	−0.116331	−	0.993210i	\(-0.537113\pi\)
							−0.727538	+	0.686067i	\(0.759336\pi\)
\(3\)	11.5896	+	10.4250i	0.743475	+	0.668764i
							\(5\)	−2.77896	+	15.7603i	−0.0497116	+	0.281929i	−0.999523	−	0.0308958i	\(-0.990164\pi\)
0.949811	+	0.312824i	\(0.101275\pi\)
\(7\)	−95.3445	+	80.0035i	−0.735446	+	0.617112i	−0.931610	−	0.363459i	\(-0.881596\pi\)
							0.196165	+	0.980571i	\(0.437151\pi\)
\(11\)	98.9514	+	561.181i	0.246570	+	1.39837i	0.816818	+	0.576896i	\(0.195736\pi\)
							−0.570248	+	0.821473i	\(0.693153\pi\)
\(13\)	−475.626	+	173.114i	−0.780562	+	0.284101i	−0.701407	−	0.712761i	\(-0.747444\pi\)
							−0.0791548	+	0.996862i	\(0.525222\pi\)
\(17\)	401.110	−	694.743i	0.336621	−	0.583045i	−0.647174	−	0.762342i	\(-0.724049\pi\)
							0.983795	+	0.179298i	\(0.0573826\pi\)
\(19\)	−992.540	−	1719.13i	−0.630760	−	1.09251i	−0.987397	−	0.158265i	\(-0.949410\pi\)
							0.356637	−	0.934243i	\(-0.383923\pi\)
\(23\)	2011.74	+	1688.05i	0.792961	+	0.665373i	0.946476	−	0.322773i	\(-0.104615\pi\)
							−0.153516	+	0.988146i	\(0.549060\pi\)
\(29\)	−2690.88	−	979.399i	−0.594154	−	0.216254i	0.0274016	−	0.999625i	\(-0.491277\pi\)
							−0.621555	+	0.783370i	\(0.713499\pi\)
\(31\)	−813.167	−	682.329i	−0.151976	−	0.127523i	0.563629	−	0.826028i	\(-0.309405\pi\)
							−0.715605	+	0.698505i	\(0.753849\pi\)
\(37\)	5032.18	−	8715.99i	0.604299	−	1.04668i	−0.387863	−	0.921717i	\(-0.626787\pi\)
							0.992162	−	0.124959i	\(-0.0398800\pi\)
\(41\)	7762.35	−	2825.26i	0.721163	−	0.262482i	0.0447437	−	0.998999i	\(-0.485753\pi\)
							0.676419	+	0.736517i	\(0.263531\pi\)
\(43\)	2897.00	+	16429.7i	0.238934	+	1.35506i	0.834168	+	0.551510i	\(0.185948\pi\)
							−0.595235	+	0.803552i	\(0.702941\pi\)
\(47\)	−13414.2	+	11255.8i	−0.885767	+	0.743247i	−0.967357	−	0.253419i	\(-0.918445\pi\)
							0.0815894	+	0.996666i	\(0.474000\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.4.5	✓	84
3.2	odd	2		81.6.e.a.64.10		84
27.7	even	9	inner	27.6.e.a.7.5	yes	84
27.13	even	9		729.6.a.c.1.31		42
27.14	odd	18		729.6.a.e.1.12		42
27.20	odd	18		81.6.e.a.19.10		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.5	✓	84	1.1	even	1	trivial
27.6.e.a.7.5	yes	84	27.7	even	9	inner
81.6.e.a.19.10		84	27.20	odd	18
81.6.e.a.64.10		84	3.2	odd	2
729.6.a.c.1.31		42	27.13	even	9
729.6.a.e.1.12		42	27.14	odd	18