Embedded newform 18.9.d.a.11.7

• Label: 18.9.d.a.11.7
• Level: $18$
• Weight: $9$
• Character: 18.11
• 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			11.7
Root			\(0.500000 + 22.4281i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.11
Dual form			18.9.d.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.79796 + 5.65685i) q^{2} +(36.7596 + 72.1785i) q^{3} +(64.0000 + 110.851i) q^{4} +(-944.709 + 545.428i) q^{5} +(-48.1341 + 915.145i) q^{6} +(1250.70 - 2166.27i) q^{7} +1448.15i q^{8} +(-3858.46 + 5306.50i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 126 q^{3} + 1024 q^{4} - 882 q^{5} + 384 q^{6} - 1846 q^{7} - 28662 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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
							\(3\)	36.7596	+	72.1785i	0.453822	+	0.891092i
\(5\)	−944.709	+	545.428i	−1.51153	+	0.872685i								−0.511625	+	0.859209i	\(0.670956\pi\)
							−0.999909	+	0.0134763i	\(0.995710\pi\)
\(7\)	1250.70	−	2166.27i	0.520906	−	0.902236i	−0.478798	−	0.877925i	\(-0.658927\pi\)
							0.999704	−	0.0243111i	\(-0.00773922\pi\)
\(11\)	−3212.80	−	1854.91i	−0.219439	−	0.126693i	0.386252	−	0.922393i	\(-0.373770\pi\)
							−0.605690	+	0.795700i	\(0.707103\pi\)
\(13\)	22409.9	+	38815.1i	0.784633	+	1.35902i	0.929218	+	0.369531i	\(0.120482\pi\)
							−0.144586	+	0.989492i	\(0.546185\pi\)
\(17\)			54543.2i			0.653048i	0.945189	+	0.326524i	\(0.105877\pi\)
							−0.945189	+	0.326524i	\(0.894123\pi\)
\(19\)	125188.			0.960616			0.480308	−	0.877100i	\(-0.340525\pi\)
							0.480308	+	0.877100i	\(0.340525\pi\)
\(23\)	266411.	−	153812.i	0.952007	−	0.549642i	0.0583033	−	0.998299i	\(-0.481431\pi\)
							0.893704	+	0.448657i	\(0.148098\pi\)
\(29\)	400230.	+	231073.i	0.565872	+	0.326706i	0.755499	−	0.655150i	\(-0.227395\pi\)
							−0.189627	+	0.981856i	\(0.560728\pi\)
\(31\)	−140570.	−	243475.i	−0.152211	−	0.263638i	0.779829	−	0.625993i	\(-0.215306\pi\)
							−0.932040	+	0.362355i	\(0.881973\pi\)
\(37\)	159985.			0.0853633			0.0426816	−	0.999089i	\(-0.486410\pi\)
							0.0426816	+	0.999089i	\(0.486410\pi\)
\(41\)	−4.49810e6	+	2.59698e6i	−1.59182	+	0.919038i	−0.598825	+	0.800880i	\(0.704365\pi\)
							−0.992995	+	0.118158i	\(0.962301\pi\)
\(43\)	2.49848e6	−	4.32750e6i	0.730807	−	1.26580i	−0.225731	−	0.974190i	\(-0.572477\pi\)
							0.956539	−	0.291606i	\(-0.0941895\pi\)
\(47\)	−306798.	−	177130.i	−0.0628725	−	0.0362995i	0.468234	−	0.883604i	\(-0.344890\pi\)
							−0.531107	+	0.847305i	\(0.678224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.9.d.a.11.7	yes	16
3.2	odd	2		54.9.d.a.35.4		16
4.3	odd	2		144.9.q.b.65.3		16
9.2	odd	6		162.9.b.c.161.16		16
9.4	even	3		54.9.d.a.17.4		16
9.5	odd	6	inner	18.9.d.a.5.7	✓	16
9.7	even	3		162.9.b.c.161.1		16
12.11	even	2		432.9.q.c.305.8		16
36.23	even	6		144.9.q.b.113.3		16
36.31	odd	6		432.9.q.c.17.8		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.9.d.a.5.7	✓	16	9.5	odd	6	inner
18.9.d.a.11.7	yes	16	1.1	even	1	trivial
54.9.d.a.17.4		16	9.4	even	3
54.9.d.a.35.4		16	3.2	odd	2
144.9.q.b.65.3		16	4.3	odd	2
144.9.q.b.113.3		16	36.23	even	6
162.9.b.c.161.1		16	9.7	even	3
162.9.b.c.161.16		16	9.2	odd	6
432.9.q.c.17.8		16	36.31	odd	6
432.9.q.c.305.8		16	12.11	even	2