Embedded newform 27.8.e.a.4.11

• Label: 27.8.e.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.493529 - 0.179630i) q^{2} +(-38.9037 + 25.9520i) q^{3} +(-97.8424 - 82.0995i) q^{4} +(-47.2754 + 268.112i) q^{5} +(23.8618 - 5.81978i) q^{6} +(932.688 - 782.618i) q^{7} +(67.1535 + 116.313i) q^{8} +(839.991 - 2019.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.493529	−	0.179630i	−0.0436222	−	0.0158772i	0.320117	−	0.947378i	\(-0.396278\pi\)
							−0.363739	+	0.931501i	\(0.618500\pi\)
\(3\)	−38.9037	+	25.9520i	−0.831891	+	0.554940i
							\(5\)	−47.2754	+	268.112i	−0.169138	+	0.959228i	0.775557	+	0.631277i	\(0.217469\pi\)
−0.944695	+	0.327950i	\(0.893642\pi\)
\(7\)	932.688	−	782.618i	1.02776	−	0.862396i	0.0371805	−	0.999309i	\(-0.488162\pi\)
							0.990583	+	0.136912i	\(0.0437179\pi\)
\(11\)	163.525	+	927.398i	0.0370434	+	0.210083i	0.997711	−	0.0676158i	\(-0.0215392\pi\)
							−0.960668	+	0.277699i	\(0.910428\pi\)
\(13\)	7907.66	−	2878.15i	0.998266	−	0.363339i	0.209350	−	0.977841i	\(-0.432865\pi\)
							0.788916	+	0.614502i	\(0.210643\pi\)
\(17\)	10208.4	−	17681.4i	0.503947	−	0.872861i	−0.496043	−	0.868298i	\(-0.665214\pi\)
							0.999990	−	0.00456334i	\(-0.00145256\pi\)
\(19\)	−23788.0	−	41202.0i	−0.795646	−	1.37810i	−0.922428	−	0.386169i	\(-0.873798\pi\)
							0.126782	−	0.991931i	\(-0.459535\pi\)
\(23\)	62547.7	+	52483.8i	1.07192	+	0.899452i	0.995225	−	0.0976027i	\(-0.0311174\pi\)
							0.0766992	+	0.997054i	\(0.475562\pi\)
\(29\)	−147074.	−	53530.6i	−1.11981	−	0.407576i	−0.285225	−	0.958460i	\(-0.592068\pi\)
							−0.834582	+	0.550884i	\(0.814291\pi\)
\(31\)	−79745.9	−	66914.7i	−0.480775	−	0.403418i	0.369931	−	0.929059i	\(-0.379381\pi\)
							−0.850707	+	0.525641i	\(0.823826\pi\)
\(37\)	108488.	−	187906.i	0.352107	−	0.609866i	−0.634512	−	0.772913i	\(-0.718799\pi\)
							0.986618	+	0.163047i	\(0.0521321\pi\)
\(41\)	708955.	−	258038.i	1.60648	−	0.584711i	0.625740	−	0.780032i	\(-0.284797\pi\)
							0.980739	+	0.195321i	\(0.0625750\pi\)
\(43\)	−51214.9	−	290454.i	−0.0982329	−	0.557106i	−0.993709	−	0.111996i	\(-0.964276\pi\)
							0.895476	−	0.445110i	\(-0.146835\pi\)
\(47\)	708465.	−	594473.i	0.995350	−	0.835198i	0.00901683	−	0.999959i	\(-0.497130\pi\)
							0.986333	+	0.164761i	\(0.0526854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.11	✓	120
3.2	odd	2		81.8.e.a.64.10		120
27.7	even	9	inner	27.8.e.a.7.11	yes	120
27.20	odd	18		81.8.e.a.19.10		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.11	✓	120	1.1	even	1	trivial
27.8.e.a.7.11	yes	120	27.7	even	9	inner
81.8.e.a.19.10		120	27.20	odd	18
81.8.e.a.64.10		120	3.2	odd	2