Embedded newform 27.8.e.a.4.13

• Label: 27.8.e.a.4.13
• Level: $27$
• Weight: $8$
• Character: 27.4
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.42719 + 2.33931i) q^{2} +(21.8338 - 41.3556i) q^{3} +(-62.2172 - 52.2064i) q^{4} +(40.2664 - 228.362i) q^{5} +(237.073 - 214.725i) q^{6} +(-1346.32 + 1129.70i) q^{7} +(-715.495 - 1239.27i) q^{8} +(-1233.57 - 1805.90i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.42719	+	2.33931i	0.568089	+	0.206768i	0.610065	−	0.792351i	\(-0.291143\pi\)
							−0.0419763	+	0.999119i	\(0.513365\pi\)
\(3\)	21.8338	−	41.3556i	0.466879	−	0.884321i
							\(5\)	40.2664	−	228.362i	0.144061	−	0.817013i	−0.824054	−	0.566511i	\(-0.808293\pi\)
0.968116	−	0.250503i	\(-0.0805959\pi\)
\(7\)	−1346.32	+	1129.70i	−1.48357	+	1.24486i	−0.581292	+	0.813695i	\(0.697452\pi\)
							−0.902274	+	0.431164i	\(0.858103\pi\)
\(11\)	151.682	+	860.231i	0.0343605	+	0.194868i	0.997156	−	0.0753622i	\(-0.0240113\pi\)
							−0.962796	+	0.270230i	\(0.912900\pi\)
\(13\)	8115.59	−	2953.83i	1.02451	−	0.372893i	0.225525	−	0.974237i	\(-0.427590\pi\)
							0.798990	+	0.601344i	\(0.205368\pi\)
\(17\)	19068.2	−	33027.1i	0.941322	−	1.63042i	0.178370	−	0.983963i	\(-0.442918\pi\)
							0.762952	−	0.646455i	\(-0.223749\pi\)
\(19\)	−14542.6	−	25188.5i	−0.486412	−	0.842490i	0.513466	−	0.858110i	\(-0.328361\pi\)
							−0.999878	+	0.0156197i	\(0.995028\pi\)
\(23\)	−228.761	−	191.954i	−0.00392045	−	0.00328964i	0.640825	−	0.767687i	\(-0.278592\pi\)
							−0.644746	+	0.764397i	\(0.723037\pi\)
\(29\)	125276.	+	45596.8i	0.953841	+	0.347170i	0.771617	−	0.636087i	\(-0.219448\pi\)
							0.182224	+	0.983257i	\(0.441670\pi\)
\(31\)	−57472.9	−	48225.5i	−0.346495	−	0.290744i	0.452886	−	0.891569i	\(-0.350395\pi\)
							−0.799381	+	0.600825i	\(0.794839\pi\)
\(37\)	−19233.6	+	33313.5i	−0.0624243	+	0.108122i	−0.895549	−	0.444964i	\(-0.853217\pi\)
							0.833124	+	0.553086i	\(0.186550\pi\)
\(41\)	134487.	−	48949.4i	0.304746	−	0.110918i	−0.185121	−	0.982716i	\(-0.559268\pi\)
							0.489867	+	0.871797i	\(0.337045\pi\)
\(43\)	−81314.8	−	461159.i	−0.155966	−	0.884527i	−0.957897	−	0.287111i	\(-0.907305\pi\)
							0.801931	−	0.597416i	\(-0.203806\pi\)
\(47\)	−778592.	+	653317.i	−1.09387	+	0.917870i	−0.996998	−	0.0774264i	\(-0.975330\pi\)
							−0.0968768	+	0.995296i	\(0.530885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.13	✓	120
3.2	odd	2		81.8.e.a.64.8		120
27.7	even	9	inner	27.8.e.a.7.13	yes	120
27.20	odd	18		81.8.e.a.19.8		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.13	✓	120	1.1	even	1	trivial
27.8.e.a.7.13	yes	120	27.7	even	9	inner
81.8.e.a.19.8		120	27.20	odd	18
81.8.e.a.64.8		120	3.2	odd	2