Embedded newform 27.7.f.a.2.17

• Label: 27.7.f.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.6025 + 2.57482i) q^{2} +(19.5272 - 18.6464i) q^{3} +(146.463 + 53.3083i) q^{4} +(-47.4564 - 56.5563i) q^{5} +(333.158 - 222.005i) q^{6} +(-559.698 + 203.713i) q^{7} +(1179.64 + 681.064i) q^{8} +(33.6255 - 728.224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.6025	+	2.57482i	1.82531	+	0.321852i	0.977899	−	0.209078i	\(-0.0670462\pi\)
							0.847416	+	0.530930i	\(0.178157\pi\)
\(3\)	19.5272	−	18.6464i	0.723231	−	0.690606i
							\(5\)	−47.4564	−	56.5563i	−0.379651	−	0.452451i	0.542053	−	0.840344i	\(-0.317647\pi\)
−0.921704	+	0.387894i	\(0.873203\pi\)
\(7\)	−559.698	+	203.713i	−1.63177	+	0.593916i	−0.985573	−	0.169252i	\(-0.945865\pi\)
							−0.646199	+	0.763169i	\(0.723643\pi\)
\(11\)	−103.550	+	123.406i	−0.0777984	+	0.0927166i	−0.803540	−	0.595251i	\(-0.797053\pi\)
							0.725741	+	0.687968i	\(0.241497\pi\)
\(13\)	611.857	+	3470.01i	0.278497	+	1.57943i	0.727631	+	0.685969i	\(0.240621\pi\)
							−0.449134	+	0.893464i	\(0.648268\pi\)
\(17\)	991.661	−	572.536i	0.201844	−	0.116535i	−0.395671	−	0.918392i	\(-0.629488\pi\)
							0.597515	+	0.801857i	\(0.296155\pi\)
\(19\)	1966.90	−	3406.77i	0.286761	−	0.496685i	−0.686273	−	0.727344i	\(-0.740755\pi\)
							0.973035	+	0.230658i	\(0.0740880\pi\)
\(23\)	2407.96	−	6615.81i	0.197909	−	0.543750i	−0.800549	−	0.599268i	\(-0.795459\pi\)
							0.998458	+	0.0555173i	\(0.0176808\pi\)
\(29\)	−17217.1	−	3035.84i	−0.705938	−	0.124476i	−0.190857	−	0.981618i	\(-0.561127\pi\)
							−0.515080	+	0.857142i	\(0.672238\pi\)
\(31\)	−11762.6	−	4281.22i	−0.394836	−	0.143708i	0.136969	−	0.990575i	\(-0.456264\pi\)
							−0.531805	+	0.846867i	\(0.678486\pi\)
\(37\)	2070.19	+	3585.67i	0.0408700	+	0.0707889i	0.885737	−	0.464188i	\(-0.153654\pi\)
							−0.844867	+	0.534977i	\(0.820320\pi\)
\(41\)	87069.7	−	15352.7i	1.26333	−	0.222758i	0.498440	−	0.866924i	\(-0.333906\pi\)
							0.764886	+	0.644166i	\(0.222795\pi\)
\(43\)	−31361.3	−	26315.3i	−0.394448	−	0.330981i	0.423895	−	0.905711i	\(-0.360662\pi\)
							−0.818343	+	0.574730i	\(0.805107\pi\)
\(47\)	41464.3	+	113922.i	0.399375	+	1.09727i	0.962590	+	0.270964i	\(0.0873424\pi\)
							−0.563214	+	0.826311i	\(0.690435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.17	✓	102
3.2	odd	2		81.7.f.a.8.1		102
27.13	even	9		81.7.f.a.71.1		102
27.14	odd	18	inner	27.7.f.a.14.17	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.17	✓	102	1.1	even	1	trivial
27.7.f.a.14.17	yes	102	27.14	odd	18	inner
81.7.f.a.8.1		102	3.2	odd	2
81.7.f.a.71.1		102	27.13	even	9