Embedded newform 27.8.e.a.16.12

• Label: 27.8.e.a.16.12
• Level: $27$
• Weight: $8$
• Character: 27.16
• 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			16.12
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47254 + 1.23561i) q^{2} +(-34.9519 - 31.0704i) q^{3} +(-21.5853 - 122.416i) q^{4} +(24.5158 + 8.92304i) q^{5} +(-13.0772 - 88.9391i) q^{6} +(-200.199 + 1135.38i) q^{7} +(242.498 - 420.019i) q^{8} +(256.266 + 2171.93i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47254	+	1.23561i	0.130155	+	0.109213i	0.705542	−	0.708669i	\(-0.250704\pi\)
							−0.575386	+	0.817882i	\(0.695148\pi\)
\(3\)	−34.9519	−	31.0704i	−0.747388	−	0.664388i
							\(5\)	24.5158	+	8.92304i	0.0877106	+	0.0319240i	0.385503	−	0.922707i	\(-0.374028\pi\)
−0.297792	+	0.954631i	\(0.596250\pi\)
\(7\)	−200.199	+	1135.38i	−0.220606	+	1.25112i	0.650302	+	0.759676i	\(0.274642\pi\)
							−0.870909	+	0.491445i	\(0.836469\pi\)
\(11\)	−7653.95	+	2785.81i	−1.73385	+	0.631070i	−0.998893	−	0.0470444i	\(-0.985020\pi\)
							−0.734957	+	0.678114i	\(0.762798\pi\)
\(13\)	6456.54	−	5417.68i	0.815076	−	0.683930i	−0.136737	−	0.990607i	\(-0.543662\pi\)
							0.951814	+	0.306677i	\(0.0992172\pi\)
\(17\)	467.099	+	809.039i	0.0230589	+	0.0399391i	0.877325	−	0.479898i	\(-0.159326\pi\)
							−0.854266	+	0.519837i	\(0.825993\pi\)
\(19\)	−13833.5	+	23960.3i	−0.462694	+	0.801410i	−0.999094	−	0.0425538i	\(-0.986451\pi\)
							0.536400	+	0.843964i	\(0.319784\pi\)
\(23\)	14051.7	+	79691.2i	0.240814	+	1.36572i	0.830017	+	0.557739i	\(0.188331\pi\)
							−0.589203	+	0.807985i	\(0.700558\pi\)
\(29\)	−31804.3	−	26687.0i	−0.242154	−	0.203192i	0.513631	−	0.858011i	\(-0.328300\pi\)
							−0.755785	+	0.654820i	\(0.772745\pi\)
\(31\)	−35874.0	−	203452.i	−0.216279	−	1.22658i	−0.878674	−	0.477423i	\(-0.841571\pi\)
							0.662395	−	0.749155i	\(-0.269540\pi\)
\(37\)	−199236.	−	345086.i	−0.646638	−	1.12001i	−0.983921	−	0.178606i	\(-0.942841\pi\)
							0.337283	−	0.941403i	\(-0.390492\pi\)
\(41\)	−332743.	+	279205.i	−0.753990	+	0.632673i	−0.936555	−	0.350521i	\(-0.886005\pi\)
							0.182565	+	0.983194i	\(0.441560\pi\)
\(43\)	−190433.	+	69311.9i	−0.365260	+	0.132944i	−0.518129	−	0.855303i	\(-0.673371\pi\)
							0.152868	+	0.988247i	\(0.451149\pi\)
\(47\)	−66681.6	+	378170.i	−0.0936836	+	0.531306i	0.901459	+	0.432864i	\(0.142497\pi\)
							−0.995143	+	0.0984420i	\(0.968614\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.12	✓	120
3.2	odd	2		81.8.e.a.46.9		120
27.5	odd	18		81.8.e.a.37.9		120
27.22	even	9	inner	27.8.e.a.22.12	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.12	✓	120	1.1	even	1	trivial
27.8.e.a.22.12	yes	120	27.22	even	9	inner
81.8.e.a.37.9		120	27.5	odd	18
81.8.e.a.46.9		120	3.2	odd	2