Embedded newform 27.8.e.a.4.5

• Label: 27.8.e.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.9790 - 5.08793i) q^{2} +(-44.6310 + 13.9670i) q^{3} +(71.4707 + 59.9711i) q^{4} +(-48.7897 + 276.700i) q^{5} +(694.958 + 31.8356i) q^{6} +(-1113.53 + 934.359i) q^{7} +(258.112 + 447.063i) q^{8} +(1796.85 - 1246.72i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.9790	−	5.08793i	−1.23558	−	0.449713i	−0.360073	−	0.932924i	\(-0.617248\pi\)
							−0.875504	+	0.483211i	\(0.839471\pi\)
\(3\)	−44.6310	+	13.9670i	−0.954360	+	0.298660i
							\(5\)	−48.7897	+	276.700i	−0.174555	+	0.989952i	0.764101	+	0.645097i	\(0.223183\pi\)
−0.938656	+	0.344855i	\(0.887928\pi\)
\(7\)	−1113.53	+	934.359i	−1.22704	+	1.02960i	−0.228609	+	0.973518i	\(0.573418\pi\)
							−0.998426	+	0.0560867i	\(0.982138\pi\)
\(11\)	−907.590	−	5147.20i	−0.205596	−	1.16599i	−0.896499	−	0.443047i	\(-0.853898\pi\)
							0.690902	−	0.722948i	\(-0.257213\pi\)
\(13\)	−12245.3	+	4456.91i	−1.54585	+	0.562642i	−0.967439	−	0.253105i	\(-0.918548\pi\)
							−0.578409	+	0.815747i	\(0.696326\pi\)
\(17\)	9922.78	−	17186.8i	0.489849	−	0.848443i	−0.510083	−	0.860125i	\(-0.670385\pi\)
							0.999932	+	0.0116820i	\(0.00371858\pi\)
\(19\)	16321.4	+	28269.5i	0.545909	+	0.945542i	0.998549	+	0.0538488i	\(0.0171489\pi\)
							−0.452640	+	0.891693i	\(0.649518\pi\)
\(23\)	−892.868	−	749.205i	−0.0153017	−	0.0128397i	0.635105	−	0.772426i	\(-0.280957\pi\)
							−0.650406	+	0.759586i	\(0.725401\pi\)
\(29\)	114798.	+	41783.0i	0.874060	+	0.318132i	0.739810	−	0.672816i	\(-0.234915\pi\)
							0.134250	+	0.990948i	\(0.457138\pi\)
\(31\)	−135317.	−	113544.i	−0.815804	−	0.684541i	0.136182	−	0.990684i	\(-0.456517\pi\)
							−0.951985	+	0.306143i	\(0.900961\pi\)
\(37\)	209464.	−	362802.i	0.679834	−	1.17751i	−0.295196	−	0.955437i	\(-0.595385\pi\)
							0.975031	−	0.222071i	\(-0.0712817\pi\)
\(41\)	215118.	−	78296.4i	0.487453	−	0.177418i	−0.0865894	−	0.996244i	\(-0.527597\pi\)
							0.574042	+	0.818826i	\(0.305375\pi\)
\(43\)	−21996.2	−	124747.i	−0.0421899	−	0.239271i	0.956419	−	0.291998i	\(-0.0943199\pi\)
							−0.998609	+	0.0527265i	\(0.983209\pi\)
\(47\)	−325194.	+	272870.i	−0.456877	+	0.383366i	−0.841981	−	0.539508i	\(-0.818610\pi\)
							0.385103	+	0.922874i	\(0.374166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.5	✓	120
3.2	odd	2		81.8.e.a.64.16		120
27.7	even	9	inner	27.8.e.a.7.5	yes	120
27.20	odd	18		81.8.e.a.19.16		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.5	✓	120	1.1	even	1	trivial
27.8.e.a.7.5	yes	120	27.7	even	9	inner
81.8.e.a.19.16		120	27.20	odd	18
81.8.e.a.64.16		120	3.2	odd	2