Embedded newform 27.9.f.a.2.5

• Label: 27.9.f.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	27.2
Dual form			27.9.f.a.14.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.5826 - 3.80559i) q^{2} +(-80.3561 - 10.1929i) q^{3} +(210.764 + 76.7119i) q^{4} +(129.757 + 154.639i) q^{5} +(1695.50 + 525.792i) q^{6} +(1479.28 - 538.412i) q^{7} +(601.827 + 347.465i) q^{8} +(6353.21 + 1638.13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.5826	−	3.80559i	−1.34891	−	0.237850i	−0.547923	−	0.836529i	\(-0.684581\pi\)
							−0.800989	+	0.598679i	\(0.795692\pi\)
\(3\)	−80.3561	−	10.1929i	−0.992051	−	0.125839i
							\(5\)	129.757	+	154.639i	0.207612	+	0.247422i	0.859795	−	0.510639i	\(-0.170591\pi\)
−0.652183	+	0.758061i	\(0.726147\pi\)
\(7\)	1479.28	−	538.412i	0.616108	−	0.224245i	−0.0150656	−	0.999887i	\(-0.504796\pi\)
							0.631174	+	0.775641i	\(0.282573\pi\)
\(11\)	−5647.40	+	6730.30i	−0.385725	+	0.459689i	−0.923612	−	0.383328i	\(-0.874778\pi\)
							0.537888	+	0.843017i	\(0.319222\pi\)
\(13\)	−2496.75	−	14159.8i	−0.0874182	−	0.495773i	−0.996809	−	0.0798295i	\(-0.974562\pi\)
							0.909390	−	0.415944i	\(-0.136549\pi\)
\(17\)	−48327.4	+	27901.8i	−0.578626	+	0.334070i	−0.760587	−	0.649236i	\(-0.775089\pi\)
							0.181961	+	0.983306i	\(0.441755\pi\)
\(19\)	24124.8	−	41785.3i	0.185118	−	0.320634i	−0.758498	−	0.651675i	\(-0.774067\pi\)
							0.943616	+	0.331041i	\(0.107400\pi\)
\(23\)	−57310.9	+	157460.i	−0.204798	+	0.562678i	−0.998987	−	0.0449922i	\(-0.985674\pi\)
							0.794189	+	0.607671i	\(0.207896\pi\)
\(29\)	−52796.6	−	9309.47i	−0.0746473	−	0.0131623i	0.136200	−	0.990681i	\(-0.456511\pi\)
							−0.210847	+	0.977519i	\(0.567622\pi\)
\(31\)	971027.	+	353425.i	1.05144	+	0.382693i	0.809206	−	0.587525i	\(-0.199898\pi\)
							0.242234	+	0.970218i	\(0.422120\pi\)
\(37\)	−932654.	−	1.61540e6i	−0.497638	−	0.861935i	0.502358	−	0.864660i	\(-0.332466\pi\)
							−0.999996	+	0.00272492i	\(0.999133\pi\)
\(41\)	−4.40599e6	+	776896.i	−1.55922	+	0.274933i	−0.885709	−	0.464241i	\(-0.846327\pi\)
							−0.673515	+	0.739174i	\(0.735216\pi\)
\(43\)	−3.96290e6	−	3.32527e6i	−1.15915	−	0.972641i	−0.159255	−	0.987238i	\(-0.550909\pi\)
							−0.999893	+	0.0145968i	\(0.995354\pi\)
\(47\)	−1.82983e6	−	5.02741e6i	−0.374990	−	1.03028i	−0.973405	−	0.229090i	\(-0.926425\pi\)
							0.598416	−	0.801186i	\(-0.295797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.f.a.2.5	✓	138
3.2	odd	2		81.9.f.a.8.19		138
27.13	even	9		81.9.f.a.71.19		138
27.14	odd	18	inner	27.9.f.a.14.5	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.f.a.2.5	✓	138	1.1	even	1	trivial
27.9.f.a.14.5	yes	138	27.14	odd	18	inner
81.9.f.a.8.19		138	3.2	odd	2
81.9.f.a.71.19		138	27.13	even	9