Embedded newform 27.9.f.a.2.17

• Label: 27.9.f.a.2.17
• Level: $27$
• Weight: $9$
• Character: 27.2
• Analytic conductor: $10.999$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9992224717\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	27.2
Dual form			27.9.f.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.2805 + 2.51803i) q^{2} +(-64.8590 - 48.5213i) q^{3} +(-42.9702 - 15.6399i) q^{4} +(501.169 + 597.270i) q^{5} +(-804.038 - 856.223i) q^{6} +(-473.173 + 172.221i) q^{7} +(-3789.11 - 2187.64i) q^{8} +(1852.37 + 6294.08i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.2805	+	2.51803i	0.892529	+	0.157377i	0.601060	−	0.799204i	\(-0.294745\pi\)
							0.291468	+	0.956580i	\(0.405856\pi\)
\(3\)	−64.8590	−	48.5213i	−0.800728	−	0.599028i
							\(5\)	501.169	+	597.270i	0.801870	+	0.955632i	0.999697	−	0.0246050i	\(-0.00783282\pi\)
−0.197827	+	0.980237i	\(0.563388\pi\)
\(7\)	−473.173	+	172.221i	−0.197073	+	0.0717289i	−0.438671	−	0.898648i	\(-0.644551\pi\)
							0.241598	+	0.970376i	\(0.422329\pi\)
\(11\)	−14064.3	+	16761.1i	−0.960608	+	1.14481i	0.0287915	+	0.999585i	\(0.490834\pi\)
							−0.989399	+	0.145222i	\(0.953610\pi\)
\(13\)	8434.04	+	47831.8i	0.295299	+	1.67472i	0.665984	+	0.745966i	\(0.268012\pi\)
							−0.370685	+	0.928759i	\(0.620877\pi\)
\(17\)	−37714.6	+	21774.5i	−0.451559	+	0.260707i	−0.708488	−	0.705723i	\(-0.750622\pi\)
							0.256930	+	0.966430i	\(0.417289\pi\)
\(19\)	26758.6	−	46347.3i	0.205328	−	0.355639i	−0.744909	−	0.667166i	\(-0.767507\pi\)
							0.950237	+	0.311527i	\(0.100840\pi\)
\(23\)	74836.0	−	205610.i	0.267423	−	0.734739i	−0.731194	−	0.682170i	\(-0.761037\pi\)
							0.998617	−	0.0525697i	\(-0.0167412\pi\)
\(29\)	−1.18416e6	−	208799.i	−1.67424	−	0.295214i	−0.745655	−	0.666332i	\(-0.767863\pi\)
							−0.928586	+	0.371118i	\(0.878975\pi\)
\(31\)	636861.	+	231798.i	0.689601	+	0.250994i	0.662964	−	0.748651i	\(-0.269298\pi\)
							0.0266367	+	0.999645i	\(0.491520\pi\)
\(37\)	−373921.	−	647649.i	−0.199514	−	0.345568i	0.748857	−	0.662731i	\(-0.230603\pi\)
							−0.948371	+	0.317164i	\(0.897270\pi\)
\(41\)	686918.	−	121122.i	0.243091	−	0.0428636i	−0.0507748	−	0.998710i	\(-0.516169\pi\)
							0.293866	+	0.955847i	\(0.405058\pi\)
\(43\)	−2.13681e6	−	1.79300e6i	−0.625017	−	0.524452i	0.274359	−	0.961627i	\(-0.411534\pi\)
							−0.899376	+	0.437176i	\(0.855979\pi\)
\(47\)	3.12827e6	+	8.59485e6i	0.641081	+	1.76136i	0.648304	+	0.761382i	\(0.275479\pi\)
							−0.00722272	+	0.999974i	\(0.502299\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.f.a.2.17	✓	138
3.2	odd	2		81.9.f.a.8.7		138
27.13	even	9		81.9.f.a.71.7		138
27.14	odd	18	inner	27.9.f.a.14.17	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.f.a.2.17	✓	138	1.1	even	1	trivial
27.9.f.a.14.17	yes	138	27.14	odd	18	inner
81.9.f.a.8.7		138	3.2	odd	2
81.9.f.a.71.7		138	27.13	even	9