Embedded newform 216.4.t.a.13.20

• Label: 216.4.t.a.13.20
• Level: $216$
• Weight: $4$
• Character: 216.13
• Analytic conductor: $12.744$
• Analytic rank: $0$
• Dimension: $636$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	216.t (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7444125612\)
Analytic rank:	\(0\)
Dimension:	\(636\)
Relative dimension:	\(106\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			13.20
Character	\(\chi\)	\(=\)	216.13
Dual form			216.4.t.a.133.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.36371 + 1.55335i) q^{2} +(4.86625 + 1.82198i) q^{3} +(3.17422 - 7.34332i) q^{4} +(0.520651 - 0.620488i) q^{5} +(-14.3326 + 3.25236i) q^{6} +(-17.5070 - 6.37202i) q^{7} +(3.90381 + 22.2881i) q^{8} +(20.3608 + 17.7324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 636 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.36371	+	1.55335i	−0.835697	+	0.549192i
							\(3\)	4.86625	+	1.82198i	0.936510	+	0.350640i
\(5\)	0.520651	−	0.620488i	0.0465684	−	0.0554981i								−0.742257	−	0.670115i	\(-0.766245\pi\)
							0.788826	+	0.614617i	\(0.210689\pi\)
\(7\)	−17.5070	−	6.37202i	−0.945288	−	0.344057i	−0.177036	−	0.984204i	\(-0.556651\pi\)
							−0.768252	+	0.640147i	\(0.778873\pi\)
\(11\)	−27.0644	−	32.2541i	−0.741839	−	0.884089i	0.254717	−	0.967016i	\(-0.418018\pi\)
							−0.996556	+	0.0829265i	\(0.973573\pi\)
\(13\)	−49.4407	−	8.71773i	−1.05480	−	0.185989i	−0.380751	−	0.924678i	\(-0.624335\pi\)
							−0.674047	+	0.738688i	\(0.735446\pi\)
\(17\)	−64.2506	+	111.285i	−0.916651	+	1.58769i	−0.112184	+	0.993687i	\(0.535785\pi\)
							−0.804466	+	0.593998i	\(0.797549\pi\)
\(19\)	−74.3063	+	42.9008i	−0.897212	+	0.518006i	−0.876295	−	0.481775i	\(-0.839992\pi\)
							−0.0209176	+	0.999781i	\(0.506659\pi\)
\(23\)	83.6232	−	30.4364i	0.758115	−	0.275931i	0.0660987	−	0.997813i	\(-0.478945\pi\)
							0.692016	+	0.721882i	\(0.256723\pi\)
\(29\)	−6.98061	+	1.23087i	−0.0446988	+	0.00788161i	−0.195953	−	0.980613i	\(-0.562780\pi\)
							0.151254	+	0.988495i	\(0.451669\pi\)
\(31\)	−210.117	+	76.4762i	−1.21736	+	0.443082i	−0.869250	−	0.494373i	\(-0.835398\pi\)
							−0.348107	+	0.937455i	\(0.613175\pi\)
\(37\)	−219.830	−	126.919i	−0.976753	−	0.563929i	−0.0754649	−	0.997148i	\(-0.524044\pi\)
							−0.901288	+	0.433220i	\(0.857377\pi\)
\(41\)	34.3334	−	194.715i	0.130780	−	0.741690i	−0.846926	−	0.531711i	\(-0.821549\pi\)
							0.977706	−	0.209979i	\(-0.0673397\pi\)
\(43\)	−22.7569	−	27.1206i	−0.0807067	−	0.0961825i	0.724180	−	0.689611i	\(-0.242218\pi\)
							−0.804887	+	0.593428i	\(0.797774\pi\)
\(47\)	53.7150	+	19.5507i	0.166705	+	0.0606757i	0.424025	−	0.905651i	\(-0.360617\pi\)
							−0.257319	+	0.966326i	\(0.582839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.4.t.a.13.20	✓	636
8.5	even	2	inner	216.4.t.a.13.98	yes	636
27.25	even	9	inner	216.4.t.a.133.98	yes	636
216.133	even	18	inner	216.4.t.a.133.20	yes	636

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.t.a.13.20	✓	636	1.1	even	1	trivial
216.4.t.a.13.98	yes	636	8.5	even	2	inner
216.4.t.a.133.20	yes	636	216.133	even	18	inner
216.4.t.a.133.98	yes	636	27.25	even	9	inner