Embedded newform 27.10.e.a.4.10

• Label: 27.10.e.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.9570 - 4.71595i) q^{2} +(-37.0275 + 135.322i) q^{3} +(-246.572 - 206.898i) q^{4} +(-222.415 + 1261.38i) q^{5} +(1117.94 - 1578.74i) q^{6} +(-4998.24 + 4194.02i) q^{7} +(5748.96 + 9957.50i) q^{8} +(-16940.9 - 10021.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.9570	−	4.71595i	−0.572623	−	0.208418i	0.0394465	−	0.999222i	\(-0.487441\pi\)
							−0.612070	+	0.790804i	\(0.709663\pi\)
\(3\)	−37.0275	+	135.322i	−0.263924	+	0.964543i
							\(5\)	−222.415	+	1261.38i	−0.159147	+	0.902568i	0.795749	+	0.605627i	\(0.207078\pi\)
−0.954896	+	0.296941i	\(0.904034\pi\)
\(7\)	−4998.24	+	4194.02i	−0.786821	+	0.660221i	−0.944956	−	0.327196i	\(-0.893896\pi\)
							0.158135	+	0.987417i	\(0.449452\pi\)
\(11\)	4387.44	+	24882.4i	0.0903533	+	0.512419i	0.996073	+	0.0885409i	\(0.0282204\pi\)
							−0.905719	+	0.423878i	\(0.860668\pi\)
\(13\)	14554.5	−	5297.40i	0.141336	−	0.0514420i	−0.270384	−	0.962753i	\(-0.587151\pi\)
							0.411719	+	0.911311i	\(0.364928\pi\)
\(17\)	187103.	−	324072.i	0.543326	−	0.941069i	−0.455384	−	0.890295i	\(-0.650498\pi\)
							0.998710	−	0.0507737i	\(-0.0161687\pi\)
\(19\)	−331545.	−	574253.i	−0.583649	−	1.01091i	−0.995042	−	0.0994513i	\(-0.968291\pi\)
							0.411394	−	0.911458i	\(-0.365042\pi\)
\(23\)	−1.82158e6	−	1.52849e6i	−1.35729	−	1.13890i	−0.976809	−	0.214114i	\(-0.931313\pi\)
							−0.380483	−	0.924788i	\(-0.624242\pi\)
\(29\)	2.27688e6	+	828718.i	0.597792	+	0.217578i	0.623153	−	0.782100i	\(-0.285851\pi\)
							−0.0253612	+	0.999678i	\(0.508074\pi\)
\(31\)	4.94152e6	+	4.14643e6i	0.961021	+	0.806392i	0.981119	−	0.193406i	\(-0.0619534\pi\)
							−0.0200981	+	0.999798i	\(0.506398\pi\)
\(37\)	−8.89215e6	+	1.54017e7i	−0.780008	+	1.35101i	0.151928	+	0.988392i	\(0.451452\pi\)
							−0.931936	+	0.362622i	\(0.881882\pi\)
\(41\)	3.98972e6	−	1.45214e6i	0.220503	−	0.0802567i	−0.229406	−	0.973331i	\(-0.573678\pi\)
							0.449910	+	0.893074i	\(0.351456\pi\)
\(43\)	−2.25330e6	−	1.27791e7i	−0.100510	−	0.570023i	−0.992919	−	0.118795i	\(-0.962097\pi\)
							0.892408	−	0.451229i	\(-0.149014\pi\)
\(47\)	−1.91163e6	+	1.60405e6i	−0.0571432	+	0.0479488i	−0.670912	−	0.741537i	\(-0.734097\pi\)
							0.613769	+	0.789486i	\(0.289653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.10	✓	156
3.2	odd	2		81.10.e.a.64.17		156
27.7	even	9	inner	27.10.e.a.7.10	yes	156
27.20	odd	18		81.10.e.a.19.17		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.10	✓	156	1.1	even	1	trivial
27.10.e.a.7.10	yes	156	27.7	even	9	inner
81.10.e.a.19.17		156	27.20	odd	18
81.10.e.a.64.17		156	3.2	odd	2