Embedded newform 18.6.c.a.7.1

• Label: 18.6.c.a.7.1
• Level: $18$
• Weight: $6$
• Character: 18.7
• Analytic conductor: $2.887$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.88690875663\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.1
Root			\(-1.22474 + 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.7
Dual form			18.6.c.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 - 3.46410i) q^{2} +(-14.6969 - 5.19615i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(-28.1969 - 48.8385i) q^{5} +(-47.3939 + 40.5194i) q^{6} +(11.1515 - 19.3150i) q^{7} -64.0000 q^{8} +(189.000 + 152.735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	−	3.46410i	0.353553	−	0.612372i
							\(3\)	−14.6969	−	5.19615i	−0.942809	−	0.333333i
\(5\)	−28.1969	−	48.8385i	−0.504402	−	0.873650i								−0.999987	−	0.00509067i	\(-0.998380\pi\)
							0.495585	−	0.868560i	\(-0.334954\pi\)
\(7\)	11.1515	−	19.3150i	0.0860180	−	0.148988i	−0.819807	−	0.572641i	\(-0.805919\pi\)
							0.905825	+	0.423653i	\(0.139252\pi\)
\(11\)	296.484	−	513.526i	0.738788	−	1.27962i	−0.214253	−	0.976778i	\(-0.568732\pi\)
							0.953041	−	0.302840i	\(-0.0979348\pi\)
\(13\)	129.531	+	224.354i	0.212576	+	0.368192i	0.952520	−	0.304476i	\(-0.0984814\pi\)
							−0.739944	+	0.672668i	\(0.765148\pi\)
\(17\)	1627.51			1.36585			0.682923	−	0.730490i	\(-0.260708\pi\)
							0.682923	+	0.730490i	\(0.260708\pi\)
\(19\)	−2642.42			−1.67926			−0.839630	−	0.543159i	\(-0.817228\pi\)
							−0.839630	+	0.543159i	\(0.817228\pi\)
\(23\)	−1568.21	−	2716.22i	−0.618138	−	1.07065i	−0.989825	−	0.142288i	\(-0.954554\pi\)
							0.371688	−	0.928358i	\(-0.378779\pi\)
\(29\)	−62.2898	+	107.889i	−0.0137538	+	0.0238222i	−0.872820	−	0.488042i	\(-0.837711\pi\)
							0.859067	+	0.511864i	\(0.171045\pi\)
\(31\)	4376.08	+	7579.60i	0.817865	+	1.41658i	0.907253	+	0.420586i	\(0.138176\pi\)
							−0.0893879	+	0.995997i	\(0.528491\pi\)
\(37\)	9741.67			1.16985			0.584924	−	0.811088i	\(-0.301124\pi\)
							0.584924	+	0.811088i	\(0.301124\pi\)
\(41\)	3735.62	+	6470.29i	0.347059	+	0.601124i	0.985726	−	0.168360i	\(-0.0538471\pi\)
							−0.638667	+	0.769484i	\(0.720514\pi\)
\(43\)	6252.46	−	10829.6i	0.515679	−	0.893183i	−0.484155	−	0.874982i	\(-0.660873\pi\)
							0.999834	−	0.0182005i	\(-0.00579373\pi\)
\(47\)	4408.73	−	7636.15i	0.291118	−	0.504231i	−0.682956	−	0.730459i	\(-0.739306\pi\)
							0.974074	+	0.226228i	\(0.0726395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.6.c.a.7.1	✓	4
3.2	odd	2		54.6.c.a.19.2		4
4.3	odd	2		144.6.i.a.97.2		4
9.2	odd	6		162.6.a.f.1.1		2
9.4	even	3	inner	18.6.c.a.13.1	yes	4
9.5	odd	6		54.6.c.a.37.2		4
9.7	even	3		162.6.a.e.1.2		2
12.11	even	2		432.6.i.a.289.2		4
36.23	even	6		432.6.i.a.145.2		4
36.31	odd	6		144.6.i.a.49.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.a.7.1	✓	4	1.1	even	1	trivial
18.6.c.a.13.1	yes	4	9.4	even	3	inner
54.6.c.a.19.2		4	3.2	odd	2
54.6.c.a.37.2		4	9.5	odd	6
144.6.i.a.49.2		4	36.31	odd	6
144.6.i.a.97.2		4	4.3	odd	2
162.6.a.e.1.2		2	9.7	even	3
162.6.a.f.1.1		2	9.2	odd	6
432.6.i.a.145.2		4	36.23	even	6
432.6.i.a.289.2		4	12.11	even	2