Embedded newform 27.8.e.a.7.13

• Label: 27.8.e.a.7.13
• Level: $27$
• Weight: $8$
• Character: 27.7
• 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			7.13
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.13

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.13	✓	120	27.4	even	9	inner
27.8.e.a.7.13	yes	120	1.1	even	1	trivial
81.8.e.a.19.8		120	3.2	odd	2
81.8.e.a.64.8		120	27.23	odd	18