Embedded newform 27.8.e.a.13.8

• Label: 27.8.e.a.13.8
• Level: $27$
• Weight: $8$
• Character: 27.13
• 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			13.8
Character	\(\chi\)	\(=\)	27.13
Dual form			27.8.e.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10962 - 6.29299i) q^{2} +(39.1042 + 25.6487i) q^{3} +(81.9102 - 29.8129i) q^{4} +(-292.309 - 245.276i) q^{5} +(118.016 - 274.543i) q^{6} +(1306.69 + 475.598i) q^{7} +(-687.466 - 1190.73i) q^{8} +(871.284 + 2005.95i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10962	−	6.29299i	−0.0980778	−	0.556227i	−0.993761	−	0.111533i	\(-0.964424\pi\)
							0.895683	−	0.444693i	\(-0.146687\pi\)
\(3\)	39.1042	+	25.6487i	0.836180	+	0.548456i
							\(5\)	−292.309	−	245.276i	−1.04579	−	0.877526i	−0.0531498	−	0.998587i	\(-0.516926\pi\)
−0.992645	+	0.121060i	\(0.961371\pi\)
\(7\)	1306.69	+	475.598i	1.43990	+	0.524079i	0.939750	−	0.341863i	\(-0.111058\pi\)
							0.500145	+	0.865942i	\(0.333280\pi\)
\(11\)	1848.96	−	1551.46i	0.418844	−	0.351452i	−0.408879	−	0.912589i	\(-0.634080\pi\)
							0.827723	+	0.561137i	\(0.189636\pi\)
\(13\)	2489.27	−	14117.3i	0.314246	−	1.78218i	−0.262170	−	0.965022i	\(-0.584438\pi\)
							0.576416	−	0.817157i	\(-0.304451\pi\)
\(17\)	−4632.16	+	8023.13i	−0.228672	+	0.396071i	−0.957415	−	0.288716i	\(-0.906772\pi\)
							0.728743	+	0.684787i	\(0.240105\pi\)
\(19\)	6624.82	+	11474.5i	0.221583	+	0.383793i	0.955289	−	0.295674i	\(-0.0955443\pi\)
							−0.733706	+	0.679467i	\(0.762211\pi\)
\(23\)	25692.1	−	9351.17i	0.440304	−	0.160258i	−0.112349	−	0.993669i	\(-0.535838\pi\)
							0.552653	+	0.833411i	\(0.313615\pi\)
\(29\)	23096.5	+	130987.i	0.175855	+	0.997321i	0.937153	+	0.348919i	\(0.113451\pi\)
							−0.761298	+	0.648402i	\(0.775438\pi\)
\(31\)	−171005.	+	62240.6i	−1.03096	+	0.375239i	−0.801447	−	0.598066i	\(-0.795936\pi\)
							−0.229514	+	0.973305i	\(0.573714\pi\)
\(37\)	−176300.	+	305361.i	−0.572199	+	0.991078i	0.424141	+	0.905596i	\(0.360576\pi\)
							−0.996340	+	0.0854815i	\(0.972757\pi\)
\(41\)	−90361.4	+	512465.i	−0.204757	+	1.16124i	0.693064	+	0.720876i	\(0.256261\pi\)
							−0.897821	+	0.440361i	\(0.854851\pi\)
\(43\)	−284369.	+	238614.i	−0.545434	+	0.457673i	−0.873391	−	0.487019i	\(-0.838084\pi\)
							0.327957	+	0.944693i	\(0.393640\pi\)
\(47\)	−262055.	−	95380.3i	−0.368172	−	0.134004i	0.151307	−	0.988487i	\(-0.451652\pi\)
							−0.519479	+	0.854483i	\(0.673874\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.13.8	✓	120
3.2	odd	2		81.8.e.a.10.13		120
27.2	odd	18		81.8.e.a.73.13		120
27.25	even	9	inner	27.8.e.a.25.8	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.13.8	✓	120	1.1	even	1	trivial
27.8.e.a.25.8	yes	120	27.25	even	9	inner
81.8.e.a.10.13		120	3.2	odd	2
81.8.e.a.73.13		120	27.2	odd	18