Embedded newform 216.2.v.b.59.2

• Label: 216.2.v.b.59.2
• Level: $216$
• Weight: $2$
• Character: 216.59
• Analytic conductor: $1.725$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.72476868366\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.2
Character	\(\chi\)	\(=\)	216.59
Dual form			216.2.v.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40972 - 0.112665i) q^{2} +(0.00307742 + 1.73205i) q^{3} +(1.97461 + 0.317652i) q^{4} +(2.47648 - 0.901367i) q^{5} +(0.190803 - 2.44205i) q^{6} +(2.55949 - 0.451307i) q^{7} +(-2.74786 - 0.670270i) q^{8} +(-2.99998 + 0.0106605i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40972	−	0.112665i	−0.996822	−	0.0796663i
							\(3\)	0.00307742	+	1.73205i	0.00177675	+	0.999998i
\(5\)	2.47648	−	0.901367i	1.10752	−	0.403103i								0.277435	−	0.960744i	\(-0.410516\pi\)
							0.830082	+	0.557641i	\(0.188293\pi\)
\(7\)	2.55949	−	0.451307i	0.967396	−	0.170578i	0.332438	−	0.943125i	\(-0.392129\pi\)
							0.634958	+	0.772547i	\(0.281018\pi\)
\(11\)	0.556608	−	1.52927i	0.167824	−	0.461091i	−0.827061	−	0.562113i	\(-0.809989\pi\)
							0.994884	+	0.101021i	\(0.0322110\pi\)
\(13\)	1.88389	−	2.24514i	0.522498	−	0.622689i	−0.438672	−	0.898647i	\(-0.644551\pi\)
							0.961169	+	0.275959i	\(0.0889953\pi\)
\(17\)	−3.28845	−	1.89859i	−0.797567	−	0.460476i	0.0450525	−	0.998985i	\(-0.485654\pi\)
							−0.842620	+	0.538509i	\(0.818988\pi\)
\(19\)	4.30904	+	7.46347i	0.988561	+	1.71224i	0.624895	+	0.780708i	\(0.285142\pi\)
							0.363666	+	0.931530i	\(0.381525\pi\)
\(23\)	−1.07009	+	6.06879i	−0.223129	+	1.26543i	0.643100	+	0.765783i	\(0.277648\pi\)
							−0.866229	+	0.499647i	\(0.833463\pi\)
\(29\)	3.88174	−	3.25716i	0.720820	−	0.604840i	−0.206792	−	0.978385i	\(-0.566302\pi\)
							0.927612	+	0.373545i	\(0.121858\pi\)
\(31\)	−3.57300	−	0.630016i	−0.641729	−	0.113154i	−0.156692	−	0.987648i	\(-0.550083\pi\)
							−0.485037	+	0.874493i	\(0.661194\pi\)
\(37\)	−6.40179	−	3.69607i	−1.05245	−	0.607631i	−0.129114	−	0.991630i	\(-0.541213\pi\)
							−0.923333	+	0.383999i	\(0.874547\pi\)
\(41\)	−4.43232	+	5.28224i	−0.692212	+	0.824947i	−0.991621	−	0.129178i	\(-0.958766\pi\)
							0.299409	+	0.954125i	\(0.403211\pi\)
\(43\)	−3.77811	−	1.37512i	−0.576156	−	0.209704i	0.0374734	−	0.999298i	\(-0.488069\pi\)
							−0.613630	+	0.789594i	\(0.710291\pi\)
\(47\)	0.253807	+	1.43941i	0.0370215	+	0.209960i	0.997707	−	0.0676790i	\(-0.0215594\pi\)
							−0.960686	+	0.277639i	\(0.910448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.b.59.2	yes	192
3.2	odd	2		648.2.v.b.611.31		192
4.3	odd	2		864.2.bh.b.815.18		192
8.3	odd	2	inner	216.2.v.b.59.17	yes	192
8.5	even	2		864.2.bh.b.815.17		192
24.11	even	2		648.2.v.b.611.16		192
27.11	odd	18	inner	216.2.v.b.11.17	yes	192
27.16	even	9		648.2.v.b.35.16		192
108.11	even	18		864.2.bh.b.335.17		192
216.11	even	18	inner	216.2.v.b.11.2	✓	192
216.43	odd	18		648.2.v.b.35.31		192
216.173	odd	18		864.2.bh.b.335.18		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.b.11.2	✓	192	216.11	even	18	inner
216.2.v.b.11.17	yes	192	27.11	odd	18	inner
216.2.v.b.59.2	yes	192	1.1	even	1	trivial
216.2.v.b.59.17	yes	192	8.3	odd	2	inner
648.2.v.b.35.16		192	27.16	even	9
648.2.v.b.35.31		192	216.43	odd	18
648.2.v.b.611.16		192	24.11	even	2
648.2.v.b.611.31		192	3.2	odd	2
864.2.bh.b.335.17		192	108.11	even	18
864.2.bh.b.335.18		192	216.173	odd	18
864.2.bh.b.815.17		192	8.5	even	2
864.2.bh.b.815.18		192	4.3	odd	2