Embedded newform 27.8.e.a.4.20

• Label: 27.8.e.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(20.7458 + 7.55086i) q^{2} +(-11.1469 - 45.4175i) q^{3} +(275.320 + 231.021i) q^{4} +(74.3769 - 421.812i) q^{5} +(111.689 - 1026.39i) q^{6} +(181.714 - 152.476i) q^{7} +(2554.39 + 4424.34i) q^{8} +(-1938.49 + 1012.53i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	20.7458	+	7.55086i	1.83369	+	0.667408i	0.991807	+	0.127748i	\(0.0407750\pi\)
							0.841883	+	0.539660i	\(0.181447\pi\)
\(3\)	−11.1469	−	45.4175i	−0.238358	−	0.971177i
							\(5\)	74.3769	−	421.812i	0.266099	−	1.50912i	−0.499789	−	0.866147i	\(-0.666589\pi\)
0.765888	−	0.642974i	\(-0.222300\pi\)
\(7\)	181.714	−	152.476i	0.200237	−	0.168019i	−0.537156	−	0.843483i	\(-0.680501\pi\)
							0.737393	+	0.675464i	\(0.236057\pi\)
\(11\)	668.319	+	3790.22i	0.151394	+	0.858599i	0.962009	+	0.273019i	\(0.0880222\pi\)
							−0.810614	+	0.585580i	\(0.800867\pi\)
\(13\)	−1107.04	+	402.929i	−0.139753	+	0.0508659i	−0.410950	−	0.911658i	\(-0.634803\pi\)
							0.271197	+	0.962524i	\(0.412581\pi\)
\(17\)	−7984.49	+	13829.5i	−0.394163	+	0.682710i	−0.992994	−	0.118165i	\(-0.962299\pi\)
							0.598831	+	0.800875i	\(0.295632\pi\)
\(19\)	−10605.9	−	18369.9i	−0.354740	−	0.614427i	0.632334	−	0.774696i	\(-0.282097\pi\)
							−0.987073	+	0.160269i	\(0.948764\pi\)
\(23\)	283.404	+	237.804i	0.00485689	+	0.00407541i	0.645213	−	0.764003i	\(-0.276769\pi\)
							−0.640356	+	0.768078i	\(0.721213\pi\)
\(29\)	−122889.	−	44727.9i	−0.935664	−	0.340554i	−0.171212	−	0.985234i	\(-0.554768\pi\)
							−0.764452	+	0.644681i	\(0.776990\pi\)
\(31\)	57631.3	+	48358.4i	0.347450	+	0.291545i	0.799765	−	0.600313i	\(-0.204957\pi\)
							−0.452315	+	0.891858i	\(0.649402\pi\)
\(37\)	51150.8	−	88595.9i	0.166015	−	0.287546i	−0.771000	−	0.636835i	\(-0.780243\pi\)
							0.937015	+	0.349289i	\(0.113577\pi\)
\(41\)	−140048.	+	50973.3i	−0.317346	+	0.115505i	−0.495782	−	0.868447i	\(-0.665119\pi\)
							0.178436	+	0.983952i	\(0.442896\pi\)
\(43\)	−85790.5	−	486542.i	−0.164551	−	0.933213i	−0.949527	−	0.313687i	\(-0.898436\pi\)
							0.784976	−	0.619526i	\(-0.212675\pi\)
\(47\)	−344242.	+	288854.i	−0.483640	+	0.405822i	−0.851740	−	0.523964i	\(-0.824453\pi\)
							0.368101	+	0.929786i	\(0.380008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.20	✓	120
3.2	odd	2		81.8.e.a.64.1		120
27.7	even	9	inner	27.8.e.a.7.20	yes	120
27.20	odd	18		81.8.e.a.19.1		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.20	✓	120	1.1	even	1	trivial
27.8.e.a.7.20	yes	120	27.7	even	9	inner
81.8.e.a.19.1		120	27.20	odd	18
81.8.e.a.64.1		120	3.2	odd	2