Embedded newform 27.10.e.a.4.20

• Label: 27.10.e.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(23.7357 + 8.63908i) q^{2} +(115.700 + 79.3502i) q^{3} +(96.5341 + 81.0017i) q^{4} +(-237.448 + 1346.63i) q^{5} +(2060.71 + 2882.98i) q^{6} +(-5786.38 + 4855.35i) q^{7} +(-4874.78 - 8443.36i) q^{8} +(7090.08 + 18361.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	23.7357	+	8.63908i	1.04898	+	0.381797i	0.808277	−	0.588802i	\(-0.200400\pi\)
							0.240702	+	0.970599i	\(0.422622\pi\)
\(3\)	115.700	+	79.3502i	0.824686	+	0.565591i
							\(5\)	−237.448	+	1346.63i	−0.169904	+	0.963572i	0.773959	+	0.633235i	\(0.218273\pi\)
−0.943863	+	0.330337i	\(0.892838\pi\)
\(7\)	−5786.38	+	4855.35i	−0.910890	+	0.764328i	−0.972288	−	0.233786i	\(-0.924889\pi\)
							0.0613978	+	0.998113i	\(0.480444\pi\)
\(11\)	7489.73	+	42476.4i	0.154241	+	0.874743i	0.959477	+	0.281788i	\(0.0909275\pi\)
							−0.805236	+	0.592955i	\(0.797961\pi\)
\(13\)	84515.8	−	30761.2i	0.820716	−	0.298716i	0.102673	−	0.994715i	\(-0.467260\pi\)
							0.718043	+	0.695999i	\(0.245038\pi\)
\(17\)	97939.0	−	169635.i	0.284404	−	0.492602i	−0.688061	−	0.725653i	\(-0.741538\pi\)
							0.972464	+	0.233051i	\(0.0748710\pi\)
\(19\)	254414.	+	440658.i	0.447868	+	0.775730i	0.998247	−	0.0591845i	\(-0.0188500\pi\)
							−0.550379	+	0.834915i	\(0.685517\pi\)
\(23\)	−82173.4	−	68951.6i	−0.0612288	−	0.0513771i	0.611660	−	0.791121i	\(-0.290502\pi\)
							−0.672888	+	0.739744i	\(0.734947\pi\)
\(29\)	−3.92260e6	−	1.42771e6i	−1.02987	−	0.374843i	−0.228840	−	0.973464i	\(-0.573493\pi\)
							−0.801032	+	0.598622i	\(0.795715\pi\)
\(31\)	6.00051e6	+	5.03503e6i	1.16697	+	0.979206i	0.999977	−	0.00675186i	\(-0.00214920\pi\)
							0.166995	+	0.985958i	\(0.446594\pi\)
\(37\)	5.90946e6	−	1.02355e7i	0.518371	−	0.897844i	−0.481402	−	0.876500i	\(-0.659872\pi\)
							0.999772	−	0.0213441i	\(-0.00679454\pi\)
\(41\)	2.01033e7	−	7.31699e6i	1.11106	−	0.404394i	0.279680	−	0.960093i	\(-0.409771\pi\)
							0.831384	+	0.555699i	\(0.187549\pi\)
\(43\)	−5.67694e6	−	3.21955e7i	−0.253225	−	1.43611i	−0.800589	−	0.599214i	\(-0.795480\pi\)
							0.547364	−	0.836895i	\(-0.315631\pi\)
\(47\)	2.96618e7	−	2.48892e7i	0.886659	−	0.743995i	−0.0808780	−	0.996724i	\(-0.525772\pi\)
							0.967537	+	0.252729i	\(0.0813280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.20	✓	156
3.2	odd	2		81.10.e.a.64.7		156
27.7	even	9	inner	27.10.e.a.7.20	yes	156
27.20	odd	18		81.10.e.a.19.7		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.20	✓	156	1.1	even	1	trivial
27.10.e.a.7.20	yes	156	27.7	even	9	inner
81.10.e.a.19.7		156	27.20	odd	18
81.10.e.a.64.7		156	3.2	odd	2