Embedded newform 27.10.e.a.4.16

• Label: 27.10.e.a.4.16
• Level: $27$
• Weight: $10$
• Character: 27.4
• Analytic conductor: $13.906$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.9059675764\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.90982 + 3.24291i) q^{2} +(132.389 - 46.4332i) q^{3} +(-323.346 - 271.320i) q^{4} +(-128.517 + 728.857i) q^{5} +(1330.14 + 15.6152i) q^{6} +(4124.79 - 3461.11i) q^{7} +(-4428.39 - 7670.20i) q^{8} +(15370.9 - 12294.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.90982	+	3.24291i	0.393762	+	0.143318i	0.531310	−	0.847177i	\(-0.321700\pi\)
							−0.137548	+	0.990495i	\(0.543922\pi\)
\(3\)	132.389	−	46.4332i	0.943643	−	0.330966i
							\(5\)	−128.517	+	728.857i	−0.0919593	+	0.521527i	0.903677	+	0.428214i	\(0.140857\pi\)
−0.995637	+	0.0933135i	\(0.970254\pi\)
\(7\)	4124.79	−	3461.11i	0.649323	−	0.544846i	−0.257543	−	0.966267i	\(-0.582913\pi\)
							0.906865	+	0.421421i	\(0.138468\pi\)
\(11\)	−9240.05	−	52402.9i	−0.190286	−	1.07917i	−0.918973	−	0.394320i	\(-0.870980\pi\)
							0.728687	−	0.684847i	\(-0.240131\pi\)
\(13\)	−31284.8	+	11386.7i	−0.303800	+	0.110574i	−0.489422	−	0.872047i	\(-0.662792\pi\)
							0.185622	+	0.982621i	\(0.440570\pi\)
\(17\)	190635.	−	330189.i	0.553582	−	0.958832i	−0.444430	−	0.895813i	\(-0.646594\pi\)
							0.998012	−	0.0630187i	\(-0.0200728\pi\)
\(19\)	−352249.	−	610114.i	−0.620096	−	1.07404i	−0.989467	−	0.144756i	\(-0.953760\pi\)
							0.369372	−	0.929282i	\(-0.379573\pi\)
\(23\)	−148915.	−	124955.i	−0.110960	−	0.0931061i	0.585620	−	0.810586i	\(-0.300851\pi\)
							−0.696579	+	0.717480i	\(0.745296\pi\)
\(29\)	−46429.4	−	16898.9i	−0.0121900	−	0.00443678i	0.335918	−	0.941891i	\(-0.390954\pi\)
							−0.348108	+	0.937454i	\(0.613176\pi\)
\(31\)	3.67387e6	+	3.08274e6i	0.714490	+	0.599528i	0.925855	−	0.377879i	\(-0.123346\pi\)
							−0.211365	+	0.977407i	\(0.567791\pi\)
\(37\)	−3.22610e6	+	5.58778e6i	−0.282990	+	0.490153i	−0.972120	−	0.234485i	\(-0.924660\pi\)
							0.689130	+	0.724638i	\(0.257993\pi\)
\(41\)	1.04038e7	−	3.78668e6i	0.574996	−	0.209281i	−0.0381215	−	0.999273i	\(-0.512137\pi\)
							0.613118	+	0.789992i	\(0.289915\pi\)
\(43\)	5.70531e6	+	3.23564e7i	0.254490	+	1.44329i	0.797378	+	0.603480i	\(0.206220\pi\)
							−0.542888	+	0.839805i	\(0.682669\pi\)
\(47\)	−4.03554e7	+	3.38622e7i	−1.20632	+	1.01222i	−0.206889	+	0.978364i	\(0.566334\pi\)
							−0.999427	+	0.0338548i	\(0.989222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.16	✓	156
3.2	odd	2		81.10.e.a.64.11		156
27.7	even	9	inner	27.10.e.a.7.16	yes	156
27.20	odd	18		81.10.e.a.19.11		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.16	✓	156	1.1	even	1	trivial
27.10.e.a.7.16	yes	156	27.7	even	9	inner
81.10.e.a.19.11		156	27.20	odd	18
81.10.e.a.64.11		156	3.2	odd	2