Embedded newform 27.6.e.a.7.7

• Label: 27.6.e.a.7.7
• 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			7.7
Character	\(\chi\)	\(=\)	27.7
Dual form			27.6.e.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.752106 + 0.273744i) q^{2} +(-2.15220 + 15.4392i) q^{3} +(-24.0227 + 20.1574i) q^{4} +(-15.6402 - 88.6998i) q^{5} +(-2.60770 - 12.2010i) q^{6} +(-70.2017 - 58.9062i) q^{7} +(25.3556 - 43.9172i) q^{8} +(-233.736 - 66.4565i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.752106	+	0.273744i	−0.132955	+	0.0483916i	−0.407641	−	0.913142i	\(-0.633648\pi\)
							0.274686	+	0.961534i	\(0.411426\pi\)
\(3\)	−2.15220	+	15.4392i	−0.138064	+	0.990423i
							\(5\)	−15.6402	−	88.6998i	−0.279780	−	1.58671i	−0.723357	−	0.690474i	\(-0.757402\pi\)
0.443577	−	0.896236i	\(-0.353709\pi\)
\(7\)	−70.2017	−	58.9062i	−0.541505	−	0.454377i	0.330547	−	0.943789i	\(-0.392767\pi\)
							−0.872052	+	0.489413i	\(0.837211\pi\)
\(11\)	−109.881	+	623.164i	−0.273804	+	1.55282i	0.468932	+	0.883234i	\(0.344639\pi\)
							−0.742736	+	0.669585i	\(0.766472\pi\)
\(13\)	−498.372	−	181.393i	−0.817891	−	0.297688i	−0.101011	−	0.994885i	\(-0.532208\pi\)
							−0.716879	+	0.697197i	\(0.754430\pi\)
\(17\)	189.216	+	327.732i	0.158795	+	0.275041i	0.934434	−	0.356135i	\(-0.115906\pi\)
							−0.775639	+	0.631176i	\(0.782572\pi\)
\(19\)	−991.118	+	1716.67i	−0.629856	+	1.09094i	0.357724	+	0.933827i	\(0.383553\pi\)
							−0.987580	+	0.157115i	\(0.949781\pi\)
\(23\)	919.191	−	771.293i	0.362315	−	0.304018i	−0.443398	−	0.896325i	\(-0.646227\pi\)
							0.805713	+	0.592307i	\(0.201783\pi\)
\(29\)	−1819.58	+	662.273i	−0.401769	+	0.146232i	−0.534998	−	0.844853i	\(-0.679688\pi\)
							0.133229	+	0.991085i	\(0.457465\pi\)
\(31\)	310.008	−	260.127i	0.0579387	−	0.0486163i	−0.613358	−	0.789805i	\(-0.710182\pi\)
							0.671297	+	0.741188i	\(0.265737\pi\)
\(37\)	−6718.58	−	11636.9i	−0.806813	−	1.39744i	−0.915060	−	0.403318i	\(-0.867857\pi\)
							0.108247	−	0.994124i	\(-0.465476\pi\)
\(41\)	7365.70	+	2680.89i	0.684312	+	0.249069i	0.660698	−	0.750652i	\(-0.270260\pi\)
							0.0236142	+	0.999721i	\(0.492483\pi\)
\(43\)	−3705.26	+	21013.6i	−0.305596	+	1.73312i	0.315089	+	0.949062i	\(0.397965\pi\)
							−0.620686	+	0.784060i	\(0.713146\pi\)
\(47\)	3409.85	+	2861.20i	0.225160	+	0.188931i	0.748388	−	0.663261i	\(-0.230828\pi\)
							−0.523228	+	0.852192i	\(0.675273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.7.7	yes	84
3.2	odd	2		81.6.e.a.19.8		84
27.2	odd	18		729.6.a.e.1.21		42
27.4	even	9	inner	27.6.e.a.4.7	✓	84
27.23	odd	18		81.6.e.a.64.8		84
27.25	even	9		729.6.a.c.1.22		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.7	✓	84	27.4	even	9	inner
27.6.e.a.7.7	yes	84	1.1	even	1	trivial
81.6.e.a.19.8		84	3.2	odd	2
81.6.e.a.64.8		84	27.23	odd	18
729.6.a.c.1.22		42	27.25	even	9
729.6.a.e.1.21		42	27.2	odd	18