Embedded newform 27.10.e.a.4.2

• Label: 27.10.e.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-39.3878 - 14.3360i) q^{2} +(29.9043 - 137.072i) q^{3} +(953.662 + 800.217i) q^{4} +(27.4469 - 155.659i) q^{5} +(-3142.92 + 4970.25i) q^{6} +(1724.19 - 1446.77i) q^{7} +(-15360.3 - 26604.9i) q^{8} +(-17894.5 - 8198.08i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−39.3878	−	14.3360i	−1.74071	−	0.633567i	−0.741413	−	0.671049i	\(-0.765844\pi\)
							−0.999297	+	0.0374827i	\(0.988066\pi\)
\(3\)	29.9043	−	137.072i	0.213151	−	0.977019i
							\(5\)	27.4469	−	155.659i	0.0196394	−	0.111381i	−0.973412	−	0.229061i	\(-0.926435\pi\)
0.993052	+	0.117680i	\(0.0375457\pi\)
\(7\)	1724.19	−	1446.77i	0.271422	−	0.227750i	−0.496909	−	0.867802i	\(-0.665532\pi\)
							0.768331	+	0.640053i	\(0.221087\pi\)
\(11\)	−11576.5	−	65653.8i	−0.238403	−	1.35205i	−0.835327	−	0.549753i	\(-0.814722\pi\)
							0.596924	−	0.802298i	\(-0.296389\pi\)
\(13\)	−88137.2	+	32079.3i	−0.855883	+	0.311516i	−0.732436	−	0.680835i	\(-0.761617\pi\)
							−0.123446	+	0.992351i	\(0.539395\pi\)
\(17\)	−104455.	+	180921.i	−0.303325	+	0.525375i	−0.976887	−	0.213756i	\(-0.931430\pi\)
							0.673562	+	0.739131i	\(0.264764\pi\)
\(19\)	149380.	+	258734.i	0.262968	+	0.455473i	0.967029	−	0.254666i	\(-0.0819654\pi\)
							−0.704061	+	0.710139i	\(0.748632\pi\)
\(23\)	−1.91489e6	−	1.60678e6i	−1.42681	−	1.19724i	−0.947566	−	0.319561i	\(-0.896465\pi\)
							−0.479249	−	0.877679i	\(-0.659091\pi\)
\(29\)	6.30782e6	+	2.29586e6i	1.65611	+	0.602773i	0.989744	−	0.142855i	\(-0.0456283\pi\)
							0.666362	+	0.745628i	\(0.267851\pi\)
\(31\)	−3.32722e6	−	2.79187e6i	−0.647074	−	0.542959i	0.259108	−	0.965848i	\(-0.416572\pi\)
							−0.906181	+	0.422889i	\(0.861016\pi\)
\(37\)	−7.49482e6	+	1.29814e7i	−0.657436	+	1.13871i	0.323841	+	0.946111i	\(0.395026\pi\)
							−0.981277	+	0.192601i	\(0.938308\pi\)
\(41\)	−7.26229e6	+	2.64326e6i	−0.401371	+	0.146087i	−0.534815	−	0.844969i	\(-0.679619\pi\)
							0.133444	+	0.991056i	\(0.457396\pi\)
\(43\)	−1.61156e6	−	9.13961e6i	−0.0718850	−	0.407680i	−0.999423	−	0.0339524i	\(-0.989191\pi\)
							0.927538	−	0.373728i	\(-0.121921\pi\)
\(47\)	−4.85098e7	+	4.07046e7i	−1.45007	+	1.21675i	−0.517542	+	0.855658i	\(0.673153\pi\)
							−0.932529	+	0.361096i	\(0.882403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.2	✓	156
3.2	odd	2		81.10.e.a.64.25		156
27.7	even	9	inner	27.10.e.a.7.2	yes	156
27.20	odd	18		81.10.e.a.19.25		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.2	✓	156	1.1	even	1	trivial
27.10.e.a.7.2	yes	156	27.7	even	9	inner
81.10.e.a.19.25		156	27.20	odd	18
81.10.e.a.64.25		156	3.2	odd	2