Embedded newform 18.9.d.a.5.2

• Label: 18.9.d.a.5.2
• Level: $18$
• Weight: $9$
• Character: 18.5
• Analytic conductor: $7.333$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	18.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.33281498110\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 8*x^15 + 5476*x^14 - 38192*x^13 + 11414542*x^12 - 67991120*x^11 + 11330952892*x^10 - 56032421816*x^9 + 5575009041895*x^8 - 21964587980304*x^7 + 1366797007396500*x^6 - 4023749554164360*x^5 + 166750774096111110*x^4 - 326820812596206264*x^3 + 9443577280676906808*x^2 - 9280833855293027160*x + 190324219382219227329
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{28}\cdot 3^{22} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.2
Root			\(0.500000 - 39.6902i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.5
Dual form			18.9.d.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.79796 + 5.65685i) q^{2} +(-24.1444 - 77.3178i) q^{3} +(64.0000 - 110.851i) q^{4} +(-69.6596 - 40.2180i) q^{5} +(673.941 + 620.976i) q^{6} +(370.849 + 642.329i) q^{7} +1448.15i q^{8} +(-5395.10 + 3733.58i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 126 q^{3} + 1024 q^{4} - 882 q^{5} + 384 q^{6} - 1846 q^{7} - 28662 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.79796	+	5.65685i	−0.612372	+	0.353553i
							\(3\)	−24.1444	−	77.3178i	−0.298079	−	0.954541i
\(5\)	−69.6596	−	40.2180i	−0.111455	−	0.0643487i								0.443236	−	0.896405i	\(-0.353830\pi\)
							−0.554691	+	0.832056i	\(0.687164\pi\)
\(7\)	370.849	+	642.329i	0.154456	+	0.267526i	0.932861	−	0.360237i	\(-0.117304\pi\)
							−0.778405	+	0.627763i	\(0.783971\pi\)
\(11\)	−19777.4	+	11418.5i	−1.35083	+	0.779900i	−0.988365	−	0.152099i	\(-0.951397\pi\)
							−0.362461	+	0.931999i	\(0.618063\pi\)
\(13\)	−6499.12	+	11256.8i	−0.227552	+	0.394132i	−0.957082	−	0.289817i	\(-0.906406\pi\)
							0.729530	+	0.683949i	\(0.239739\pi\)
\(17\)			105666.i			1.26514i	0.774502	+	0.632571i	\(0.218000\pi\)
							−0.774502	+	0.632571i	\(0.782000\pi\)
\(19\)	118278.			0.907594			0.453797	−	0.891105i	\(-0.350069\pi\)
							0.453797	+	0.891105i	\(0.350069\pi\)
\(23\)	−77410.8	−	44693.1i	−0.276624	−	0.159709i	0.355270	−	0.934764i	\(-0.384389\pi\)
							−0.631894	+	0.775055i	\(0.717722\pi\)
\(29\)	−952037.	+	549659.i	−1.34605	+	0.777143i	−0.987688	−	0.156439i	\(-0.949999\pi\)
							−0.358364	+	0.933582i	\(0.616665\pi\)
\(31\)	226348.	−	392047.i	0.245093	−	0.424513i	−0.717065	−	0.697007i	\(-0.754515\pi\)
							0.962158	+	0.272493i	\(0.0878482\pi\)
\(37\)	−2.12547e6			−1.13409			−0.567046	−	0.823686i	\(-0.691914\pi\)
							−0.567046	+	0.823686i	\(0.691914\pi\)
\(41\)	−1.45174e6	−	838161.i	−0.513751	−	0.296614i	0.220623	−	0.975359i	\(-0.429191\pi\)
							−0.734374	+	0.678745i	\(0.762524\pi\)
\(43\)	−569118.	−	985742.i	−0.166467	−	0.288330i	0.770708	−	0.637188i	\(-0.219903\pi\)
							−0.937175	+	0.348859i	\(0.886569\pi\)
\(47\)	−3.10681e6	+	1.79372e6i	−0.636683	+	0.367589i	−0.783336	−	0.621599i	\(-0.786483\pi\)
							0.146653	+	0.989188i	\(0.453150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.9.d.a.5.2	✓	16
3.2	odd	2		54.9.d.a.17.6		16
4.3	odd	2		144.9.q.b.113.6		16
9.2	odd	6	inner	18.9.d.a.11.2	yes	16
9.4	even	3		162.9.b.c.161.4		16
9.5	odd	6		162.9.b.c.161.13		16
9.7	even	3		54.9.d.a.35.6		16
12.11	even	2		432.9.q.c.17.4		16
36.7	odd	6		432.9.q.c.305.4		16
36.11	even	6		144.9.q.b.65.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.9.d.a.5.2	✓	16	1.1	even	1	trivial
18.9.d.a.11.2	yes	16	9.2	odd	6	inner
54.9.d.a.17.6		16	3.2	odd	2
54.9.d.a.35.6		16	9.7	even	3
144.9.q.b.65.6		16	36.11	even	6
144.9.q.b.113.6		16	4.3	odd	2
162.9.b.c.161.4		16	9.4	even	3
162.9.b.c.161.13		16	9.5	odd	6
432.9.q.c.17.4		16	12.11	even	2
432.9.q.c.305.4		16	36.7	odd	6