Embedded newform 27.7.f.a.2.16

• Label: 27.7.f.a.2.16
• Level: $27$
• Weight: $7$
• Character: 27.2
• Analytic conductor: $6.211$
• Analytic rank: $0$
• Dimension: $102$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.21146025774\)
Analytic rank:	\(0\)
Dimension:	\(102\)
Relative dimension:	\(17\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.16
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.7878 + 2.43117i) q^{2} +(-26.9804 + 1.02752i) q^{3} +(124.053 + 45.1517i) q^{4} +(118.006 + 140.634i) q^{5} +(-374.500 - 51.4266i) q^{6} +(-10.3283 + 3.75918i) q^{7} +(824.665 + 476.120i) q^{8} +(726.888 - 55.4459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.7878	+	2.43117i	1.72348	+	0.303896i	0.945796	−	0.324762i	\(-0.105284\pi\)
							0.777682	+	0.628658i	\(0.216395\pi\)
\(3\)	−26.9804	+	1.02752i	−0.999276	+	0.0380563i
							\(5\)	118.006	+	140.634i	0.944047	+	1.12507i	0.992002	+	0.126222i	\(0.0402850\pi\)
−0.0479554	+	0.998849i	\(0.515271\pi\)
\(7\)	−10.3283	+	3.75918i	−0.0301115	+	0.0109597i	−0.357032	−	0.934092i	\(-0.616211\pi\)
							0.326920	+	0.945052i	\(0.393989\pi\)
\(11\)	−109.488	+	130.482i	−0.0822596	+	0.0980332i	−0.805604	−	0.592454i	\(-0.798159\pi\)
							0.723345	+	0.690487i	\(0.242604\pi\)
\(13\)	−535.873	−	3039.09i	−0.243911	−	1.38329i	−0.823008	−	0.568030i	\(-0.807706\pi\)
							0.579097	−	0.815259i	\(-0.303405\pi\)
\(17\)	−6691.85	+	3863.54i	−1.36207	+	0.786391i	−0.989899	−	0.141773i	\(-0.954720\pi\)
							−0.372171	+	0.928164i	\(0.621386\pi\)
\(19\)	5440.69	−	9423.54i	0.793219	−	1.37389i	−0.130746	−	0.991416i	\(-0.541737\pi\)
							0.923964	−	0.382479i	\(-0.124930\pi\)
\(23\)	2838.25	−	7798.03i	0.233274	−	0.640916i	−0.766725	−	0.641976i	\(-0.778115\pi\)
							0.999999	+	0.00105957i	\(0.000337270\pi\)
\(29\)	33308.2	+	5873.13i	1.36570	+	0.240810i	0.807977	−	0.589214i	\(-0.200563\pi\)
							0.557727	+	0.830024i	\(0.311674\pi\)
\(31\)	−13610.5	−	4953.82i	−0.456866	−	0.166286i	0.103327	−	0.994647i	\(-0.467051\pi\)
							−0.560194	+	0.828362i	\(0.689273\pi\)
\(37\)	33152.3	+	57421.5i	0.654499	+	1.13363i	0.982019	+	0.188781i	\(0.0604537\pi\)
							−0.327520	+	0.944844i	\(0.606213\pi\)
\(41\)	9969.66	−	1757.92i	0.144653	−	0.0255063i	−0.100852	−	0.994901i	\(-0.532157\pi\)
							0.245506	+	0.969395i	\(0.421046\pi\)
\(43\)	−71047.8	−	59616.2i	−0.893605	−	0.749823i	0.0753253	−	0.997159i	\(-0.476000\pi\)
							−0.968930	+	0.247336i	\(0.920445\pi\)
\(47\)	2562.80	+	7041.24i	0.0246843	+	0.0678196i	0.951423	−	0.307888i	\(-0.0996221\pi\)
							−0.926738	+	0.375707i	\(0.877400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.16	✓	102
3.2	odd	2		81.7.f.a.8.2		102
27.13	even	9		81.7.f.a.71.2		102
27.14	odd	18	inner	27.7.f.a.14.16	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.16	✓	102	1.1	even	1	trivial
27.7.f.a.14.16	yes	102	27.14	odd	18	inner
81.7.f.a.8.2		102	3.2	odd	2
81.7.f.a.71.2		102	27.13	even	9