Embedded newform 27.7.f.a.2.3

• Label: 27.7.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.6551 - 2.05510i) q^{2} +(-14.9225 + 22.5015i) q^{3} +(71.4770 + 26.0155i) q^{4} +(105.310 + 125.503i) q^{5} +(220.166 - 231.590i) q^{6} +(-278.603 + 101.403i) q^{7} +(-123.650 - 71.3896i) q^{8} +(-283.637 - 671.559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.6551	−	2.05510i	−1.45688	−	0.256888i	−0.611584	−	0.791179i	\(-0.709467\pi\)
							−0.845300	+	0.534291i	\(0.820579\pi\)
\(3\)	−14.9225	+	22.5015i	−0.552686	+	0.833390i
							\(5\)	105.310	+	125.503i	0.842477	+	1.00402i	0.999864	+	0.0164878i	\(0.00524846\pi\)
−0.157387	+	0.987537i	\(0.550307\pi\)
\(7\)	−278.603	+	101.403i	−0.812254	+	0.295636i	−0.714555	−	0.699579i	\(-0.753371\pi\)
							−0.0976997	+	0.995216i	\(0.531148\pi\)
\(11\)	−1207.08	+	1438.54i	−0.906897	+	1.08080i	0.0895004	+	0.995987i	\(0.471473\pi\)
							−0.996397	+	0.0848106i	\(0.972971\pi\)
\(13\)	−524.739	−	2975.94i	−0.238843	−	1.35455i	−0.834366	−	0.551210i	\(-0.814166\pi\)
							0.595523	−	0.803338i	\(-0.296945\pi\)
\(17\)	3385.84	−	1954.82i	0.689159	−	0.397886i	−0.114138	−	0.993465i	\(-0.536410\pi\)
							0.803297	+	0.595579i	\(0.203077\pi\)
\(19\)	−1016.82	+	1761.19i	−0.148246	+	0.256770i	−0.930579	−	0.366090i	\(-0.880696\pi\)
							0.782333	+	0.622860i	\(0.214030\pi\)
\(23\)	2009.15	−	5520.08i	0.165131	−	0.453693i	−0.829335	−	0.558751i	\(-0.811281\pi\)
							0.994466	+	0.105058i	\(0.0335029\pi\)
\(29\)	−38554.4	−	6798.17i	−1.58081	−	0.278739i	−0.686820	−	0.726828i	\(-0.740994\pi\)
							−0.893989	+	0.448088i	\(0.852105\pi\)
\(31\)	−23238.8	−	8458.24i	−0.780061	−	0.283919i	−0.0788632	−	0.996885i	\(-0.525129\pi\)
							−0.701198	+	0.712966i	\(0.747351\pi\)
\(37\)	−25350.0	−	43907.6i	−0.500465	−	0.866831i	−1.00000		0.000536843i	\(-0.999829\pi\)
							0.499535	−	0.866294i	\(-0.333504\pi\)
\(41\)	5206.04	−	917.965i	0.0755363	−	0.0133191i	−0.135752	−	0.990743i	\(-0.543345\pi\)
							0.211289	+	0.977424i	\(0.432234\pi\)
\(43\)	99078.2	+	83136.5i	1.24616	+	1.04565i	0.997017	+	0.0771829i	\(0.0245925\pi\)
							0.249140	+	0.968467i	\(0.419852\pi\)
\(47\)	−30602.8	−	84080.5i	−0.294759	−	0.809845i	−0.995354	−	0.0962852i	\(-0.969304\pi\)
							0.700594	−	0.713560i	\(-0.252918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.3	✓	102
3.2	odd	2		81.7.f.a.8.15		102
27.13	even	9		81.7.f.a.71.15		102
27.14	odd	18	inner	27.7.f.a.14.3	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.3	✓	102	1.1	even	1	trivial
27.7.f.a.14.3	yes	102	27.14	odd	18	inner
81.7.f.a.8.15		102	3.2	odd	2
81.7.f.a.71.15		102	27.13	even	9