Embedded newform 27.6.e.a.4.3

• Label: 27.6.e.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	27.4
Dual form			27.6.e.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.30121 - 2.65742i) q^{2} +(-0.728957 - 15.5714i) q^{3} +(21.7323 + 18.2356i) q^{4} +(15.2097 - 86.2586i) q^{5} +(-36.0575 + 115.627i) q^{6} +(-90.4944 + 75.9338i) q^{7} +(14.1040 + 24.4288i) q^{8} +(-241.937 + 22.7018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.30121	−	2.65742i	−1.29068	−	0.469770i	−0.396732	−	0.917934i	\(-0.629856\pi\)
							−0.893951	+	0.448164i	\(0.852078\pi\)
\(3\)	−0.728957	−	15.5714i	−0.0467626	−	0.998906i
							\(5\)	15.2097	−	86.2586i	0.272080	−	1.54304i	−0.476008	−	0.879441i	\(-0.657917\pi\)
0.748088	−	0.663599i	\(-0.230972\pi\)
\(7\)	−90.4944	+	75.9338i	−0.698034	+	0.585720i	−0.921214	−	0.389057i	\(-0.872801\pi\)
							0.223180	+	0.974777i	\(0.428356\pi\)
\(11\)	19.6881	+	111.656i	0.0490593	+	0.278229i	0.999462	−	0.0327925i	\(-0.0104400\pi\)
							−0.950403	+	0.311021i	\(0.899329\pi\)
\(13\)	−178.502	+	64.9696i	−0.292945	+	0.106623i	−0.484312	−	0.874895i	\(-0.660930\pi\)
							0.191367	+	0.981519i	\(0.438708\pi\)
\(17\)	522.735	−	905.403i	0.438691	−	0.759836i	−0.558898	−	0.829237i	\(-0.688776\pi\)
							0.997589	+	0.0694012i	\(0.0221088\pi\)
\(19\)	1157.44	+	2004.75i	0.735556	+	1.27402i	0.954479	+	0.298278i	\(0.0964123\pi\)
							−0.218923	+	0.975742i	\(0.570254\pi\)
\(23\)	−3297.53	−	2766.96i	−1.29978	−	1.09064i	−0.990183	−	0.139777i	\(-0.955361\pi\)
							−0.309597	−	0.950868i	\(-0.600194\pi\)
\(29\)	−4184.88	−	1523.17i	−0.924034	−	0.336321i	−0.164192	−	0.986428i	\(-0.552502\pi\)
							−0.759842	+	0.650108i	\(0.774724\pi\)
\(31\)	−1590.49	−	1334.58i	−0.297253	−	0.249425i	0.481947	−	0.876200i	\(-0.339930\pi\)
							−0.779200	+	0.626776i	\(0.784374\pi\)
\(37\)	5112.67	−	8855.40i	0.613964	−	1.06342i	−0.376601	−	0.926376i	\(-0.622907\pi\)
							0.990565	−	0.137042i	\(-0.0437594\pi\)
\(41\)	−10111.3	+	3680.22i	−0.939395	+	0.341912i	−0.765927	−	0.642928i	\(-0.777720\pi\)
							−0.173468	+	0.984839i	\(0.555497\pi\)
\(43\)	−2357.48	−	13369.9i	−0.194436	−	1.10270i	−0.913220	−	0.407467i	\(-0.866412\pi\)
							0.718784	−	0.695233i	\(-0.244699\pi\)
\(47\)	21534.9	−	18069.9i	1.42199	−	1.19319i	0.471732	−	0.881742i	\(-0.343629\pi\)
							0.950262	−	0.311452i	\(-0.100815\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.4.3	✓	84
3.2	odd	2		81.6.e.a.64.12		84
27.7	even	9	inner	27.6.e.a.7.3	yes	84
27.13	even	9		729.6.a.c.1.34		42
27.14	odd	18		729.6.a.e.1.9		42
27.20	odd	18		81.6.e.a.19.12		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.3	✓	84	1.1	even	1	trivial
27.6.e.a.7.3	yes	84	27.7	even	9	inner
81.6.e.a.19.12		84	27.20	odd	18
81.6.e.a.64.12		84	3.2	odd	2
729.6.a.c.1.34		42	27.13	even	9
729.6.a.e.1.9		42	27.14	odd	18