Embedded newform 27.8.e.a.4.7

• Label: 27.8.e.a.4.7
• Level: $27$
• Weight: $8$
• Character: 27.4
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.32840 - 3.39526i) q^{2} +(39.8042 + 24.5484i) q^{3} +(-22.5624 - 18.9321i) q^{4} +(12.2615 - 69.5386i) q^{5} +(-287.961 - 364.143i) q^{6} +(48.0590 - 40.3263i) q^{7} +(781.525 + 1353.64i) q^{8} +(981.750 + 1954.26i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.32840	−	3.39526i	−0.824522	−	0.300102i	−0.104913	−	0.994481i	\(-0.533457\pi\)
							−0.719609	+	0.694380i	\(0.755679\pi\)
\(3\)	39.8042	+	24.5484i	0.851147	+	0.524927i
							\(5\)	12.2615	−	69.5386i	0.0438682	−	0.248789i	−0.954986	−	0.296652i	\(-0.904130\pi\)
0.998854	+	0.0478628i	\(0.0152410\pi\)
\(7\)	48.0590	−	40.3263i	0.0529580	−	0.0444370i	−0.615924	−	0.787805i	\(-0.711217\pi\)
							0.668882	+	0.743368i	\(0.266773\pi\)
\(11\)	−1381.25	−	7833.47i	−0.312895	−	1.77451i	−0.583789	−	0.811905i	\(-0.698431\pi\)
							0.270894	−	0.962609i	\(-0.412681\pi\)
\(13\)	12860.0	−	4680.64i	1.62345	−	0.590886i	0.639412	−	0.768864i	\(-0.279178\pi\)
							0.984034	+	0.177978i	\(0.0569557\pi\)
\(17\)	15797.5	−	27362.2i	0.779863	−	1.35076i	−0.152157	−	0.988356i	\(-0.548622\pi\)
							0.932020	−	0.362406i	\(-0.118045\pi\)
\(19\)	7936.89	+	13747.1i	0.265468	+	0.459804i	0.967686	−	0.252158i	\(-0.0811401\pi\)
							−0.702218	+	0.711962i	\(0.747807\pi\)
\(23\)	−30441.1	−	25543.1i	−0.521691	−	0.437750i	0.343530	−	0.939142i	\(-0.388377\pi\)
							−0.865221	+	0.501391i	\(0.832822\pi\)
\(29\)	−16930.2	−	6162.08i	−0.128905	−	0.0469175i	0.276762	−	0.960938i	\(-0.410739\pi\)
							−0.405667	+	0.914021i	\(0.632961\pi\)
\(31\)	8473.52	+	7110.13i	0.0510855	+	0.0428659i	0.667973	−	0.744186i	\(-0.267162\pi\)
							−0.616887	+	0.787051i	\(0.711607\pi\)
\(37\)	−182052.	+	315323.i	−0.590866	+	1.02341i	0.403250	+	0.915090i	\(0.367880\pi\)
							−0.994116	+	0.108320i	\(0.965453\pi\)
\(41\)	−421310.	+	153344.i	−0.954682	+	0.347476i	−0.771947	−	0.635686i	\(-0.780717\pi\)
							−0.182735	+	0.983162i	\(0.558495\pi\)
\(43\)	−72356.8	−	410356.i	−0.138784	−	0.787083i	−0.972150	−	0.234361i	\(-0.924700\pi\)
							0.833366	−	0.552722i	\(-0.186411\pi\)
\(47\)	501279.	−	420623.i	0.704266	−	0.590949i	−0.218718	−	0.975788i	\(-0.570187\pi\)
							0.922984	+	0.384839i	\(0.125743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.7	✓	120
3.2	odd	2		81.8.e.a.64.14		120
27.7	even	9	inner	27.8.e.a.7.7	yes	120
27.20	odd	18		81.8.e.a.19.14		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.7	✓	120	1.1	even	1	trivial
27.8.e.a.7.7	yes	120	27.7	even	9	inner
81.8.e.a.19.14		120	27.20	odd	18
81.8.e.a.64.14		120	3.2	odd	2