Embedded newform 27.9.f.a.2.20

• Label: 27.9.f.a.2.20
• Level: $27$
• Weight: $9$
• Character: 27.2
• Analytic conductor: $10.999$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9992224717\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	27.2
Dual form			27.9.f.a.14.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25.1387 + 4.43262i) q^{2} +(-80.5339 - 8.67746i) q^{3} +(371.742 + 135.303i) q^{4} +(-546.503 - 651.297i) q^{5} +(-1986.05 - 575.116i) q^{6} +(-3663.86 + 1333.53i) q^{7} +(3086.08 + 1781.75i) q^{8} +(6410.40 + 1397.66i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25.1387	+	4.43262i	1.57117	+	0.277039i	0.890302	−	0.455371i	\(-0.150493\pi\)
							0.680864	+	0.732410i	\(0.261604\pi\)
\(3\)	−80.5339	−	8.67746i	−0.994245	−	0.107129i
							\(5\)	−546.503	−	651.297i	−0.874405	−	1.04208i	−0.998757	−	0.0498377i	\(-0.984130\pi\)
0.124352	−	0.992238i	\(-0.460315\pi\)
\(7\)	−3663.86	+	1333.53i	−1.52597	+	0.555408i	−0.962631	−	0.270817i	\(-0.912706\pi\)
							−0.563340	+	0.826225i	\(0.690484\pi\)
\(11\)	−7165.64	+	8539.67i	−0.489423	+	0.583271i	−0.953071	−	0.302748i	\(-0.902096\pi\)
							0.463648	+	0.886020i	\(0.346540\pi\)
\(13\)	−6824.34	−	38702.7i	−0.238939	−	1.35509i	−0.834158	−	0.551526i	\(-0.814046\pi\)
							0.595219	−	0.803564i	\(-0.297065\pi\)
\(17\)	109614.	−	63285.8i	1.31242	−	0.757723i	0.329920	−	0.944009i	\(-0.392978\pi\)
							0.982496	+	0.186286i	\(0.0596450\pi\)
\(19\)	−62328.9	+	107957.i	−0.478272	+	0.828391i	−0.999690	−	0.0249102i	\(-0.992070\pi\)
							0.521418	+	0.853302i	\(0.325403\pi\)
\(23\)	−84577.6	+	232375.i	−0.302235	+	0.830383i	0.691876	+	0.722016i	\(0.256784\pi\)
							−0.994111	+	0.108367i	\(0.965438\pi\)
\(29\)	−177273.	−	31258.0i	−0.250640	−	0.0441946i	0.0469165	−	0.998899i	\(-0.485061\pi\)
							−0.297556	+	0.954704i	\(0.596172\pi\)
\(31\)	557486.	+	202908.i	0.603653	+	0.219712i	0.625724	−	0.780045i	\(-0.284804\pi\)
							−0.0220706	+	0.999756i	\(0.507026\pi\)
\(37\)	−229672.	−	397803.i	−0.122546	−	0.212257i	0.798225	−	0.602360i	\(-0.205773\pi\)
							−0.920771	+	0.390103i	\(0.872439\pi\)
\(41\)	−981871.	+	173130.i	−0.347471	+	0.0612686i	−0.344660	−	0.938727i	\(-0.612006\pi\)
							−0.00281108	+	0.999996i	\(0.500895\pi\)
\(43\)	−2.11904e6	−	1.77809e6i	−0.619820	−	0.520091i	0.277927	−	0.960602i	\(-0.410353\pi\)
							−0.897747	+	0.440512i	\(0.854797\pi\)
\(47\)	−1.21462e6	−	3.33714e6i	−0.248914	−	0.683885i	−0.999727	−	0.0233703i	\(-0.992560\pi\)
							0.750813	−	0.660515i	\(-0.229662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.f.a.2.20	✓	138
3.2	odd	2		81.9.f.a.8.4		138
27.13	even	9		81.9.f.a.71.4		138
27.14	odd	18	inner	27.9.f.a.14.20	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.f.a.2.20	✓	138	1.1	even	1	trivial
27.9.f.a.14.20	yes	138	27.14	odd	18	inner
81.9.f.a.8.4		138	3.2	odd	2
81.9.f.a.71.4		138	27.13	even	9