Embedded newform 27.10.e.a.4.13

• Label: 27.10.e.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.57613 - 1.66557i) q^{2} +(-24.3571 - 138.166i) q^{3} +(-374.048 - 313.863i) q^{4} +(461.629 - 2618.03i) q^{5} +(-118.664 + 672.832i) q^{6} +(5320.32 - 4464.28i) q^{7} +(2435.60 + 4218.58i) q^{8} +(-18496.5 + 6730.62i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.57613	−	1.66557i	−0.202238	−	0.0736087i	0.238915	−	0.971040i	\(-0.423208\pi\)
							−0.441153	+	0.897432i	\(0.645430\pi\)
\(3\)	−24.3571	−	138.166i	−0.173612	−	0.984814i
							\(5\)	461.629	−	2618.03i	0.330315	−	1.87331i	−0.139024	−	0.990289i	\(-0.544396\pi\)
0.469338	−	0.883018i	\(-0.344492\pi\)
\(7\)	5320.32	−	4464.28i	0.837523	−	0.702765i	−0.119482	−	0.992836i	\(-0.538124\pi\)
							0.957005	+	0.290071i	\(0.0936791\pi\)
\(11\)	−5911.61	−	33526.4i	−0.121741	−	0.690430i	−0.983190	−	0.182585i	\(-0.941553\pi\)
							0.861449	−	0.507845i	\(-0.169558\pi\)
\(13\)	115823.	−	42156.3i	1.12474	−	0.409371i	0.288359	−	0.957522i	\(-0.406890\pi\)
							0.836379	+	0.548151i	\(0.184668\pi\)
\(17\)	54853.3	−	95008.6i	0.159288	−	0.275894i	−0.775324	−	0.631563i	\(-0.782414\pi\)
							0.934612	+	0.355669i	\(0.115747\pi\)
\(19\)	−2962.69	−	5131.52i	−0.00521548	−	0.00903348i	0.863406	−	0.504510i	\(-0.168327\pi\)
							−0.868621	+	0.495476i	\(0.834993\pi\)
\(23\)	140728.	+	118085.i	0.104859	+	0.0879871i	0.693710	−	0.720255i	\(-0.255975\pi\)
							−0.588851	+	0.808242i	\(0.700419\pi\)
\(29\)	5.64337e6	+	2.05402e6i	1.48166	+	0.539278i	0.951238	−	0.308458i	\(-0.0998130\pi\)
							0.530417	+	0.847737i	\(0.322035\pi\)
\(31\)	6.88793e6	+	5.77966e6i	1.33956	+	1.12402i	0.981740	+	0.190230i	\(0.0609235\pi\)
							0.357817	+	0.933792i	\(0.383521\pi\)
\(37\)	−2.83524e6	+	4.91077e6i	−0.248703	+	0.430767i	−0.963166	−	0.268907i	\(-0.913338\pi\)
							0.714463	+	0.699673i	\(0.246671\pi\)
\(41\)	3.13238e7	−	1.14009e7i	1.73120	−	0.630105i	0.732484	−	0.680785i	\(-0.238361\pi\)
							0.998715	+	0.0506797i	\(0.0161388\pi\)
\(43\)	−1.71312e6	−	9.71560e6i	−0.0764153	−	0.433373i	−0.998881	−	0.0472937i	\(-0.984940\pi\)
							0.922466	−	0.386079i	\(-0.126171\pi\)
\(47\)	2.21279e7	−	1.85675e7i	0.661455	−	0.555027i	−0.249068	−	0.968486i	\(-0.580124\pi\)
							0.910522	+	0.413460i	\(0.135680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.13	✓	156
3.2	odd	2		81.10.e.a.64.14		156
27.7	even	9	inner	27.10.e.a.7.13	yes	156
27.20	odd	18		81.10.e.a.19.14		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.13	✓	156	1.1	even	1	trivial
27.10.e.a.7.13	yes	156	27.7	even	9	inner
81.10.e.a.19.14		156	27.20	odd	18
81.10.e.a.64.14		156	3.2	odd	2