Embedded newform 18.7.d.a.11.1

• Label: 18.7.d.a.11.1
• Level: $18$
• Weight: $7$
• Character: 18.11
• Analytic conductor: $4.141$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	18.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.14097350516\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.1
Root			\(-4.28281i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.11
Dual form			18.7.d.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.89898 - 2.82843i) q^{2} +(-22.5447 + 14.8572i) q^{3} +(16.0000 + 27.7128i) q^{4} +(156.951 - 90.6160i) q^{5} +(152.469 - 9.01919i) q^{6} +(104.306 - 180.663i) q^{7} -181.019i q^{8} +(287.526 - 669.903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.89898	−	2.82843i	−0.612372	−	0.353553i
							\(3\)	−22.5447	+	14.8572i	−0.834989	+	0.550267i
\(5\)	156.951	−	90.6160i	1.25561	−	0.724928i								0.283394	−	0.959004i	\(-0.408540\pi\)
							0.972218	+	0.234076i	\(0.0752064\pi\)
\(7\)	104.306	−	180.663i	0.304099	−	0.526715i	−0.672961	−	0.739678i	\(-0.734978\pi\)
							0.977060	+	0.212963i	\(0.0683113\pi\)
\(11\)	2301.99	+	1329.05i	1.72952	+	0.998538i	0.891782	+	0.452465i	\(0.149455\pi\)
							0.837737	+	0.546074i	\(0.183878\pi\)
\(13\)	−438.599	−	759.676i	−0.199636	−	0.345779i	0.748775	−	0.662825i	\(-0.230642\pi\)
							−0.948410	+	0.317046i	\(0.897309\pi\)
\(17\)		−	4428.53i		−	0.901389i	−0.892678	−	0.450695i	\(-0.851176\pi\)
							0.892678	−	0.450695i	\(-0.148824\pi\)
\(19\)	−4194.21			−0.611491			−0.305745	−	0.952113i	\(-0.598906\pi\)
							−0.305745	+	0.952113i	\(0.598906\pi\)
\(23\)	10399.9	−	6004.40i	0.854766	−	0.493499i	−0.00749035	−	0.999972i	\(-0.502384\pi\)
							0.862256	+	0.506473i	\(0.169051\pi\)
\(29\)	2288.79	+	1321.43i	0.0938450	+	0.0541815i	0.546188	−	0.837663i	\(-0.316078\pi\)
							−0.452343	+	0.891844i	\(0.649412\pi\)
\(31\)	−2902.50	−	5027.28i	−0.0974289	−	0.168752i	0.813191	−	0.581997i	\(-0.197729\pi\)
							−0.910620	+	0.413245i	\(0.864395\pi\)
\(37\)	41579.5			0.820869			0.410435	−	0.911890i	\(-0.365377\pi\)
							0.410435	+	0.911890i	\(0.365377\pi\)
\(41\)	−64721.2	+	37366.8i	−0.939063	+	0.542168i	−0.889666	−	0.456611i	\(-0.849063\pi\)
							−0.0493965	+	0.998779i	\(0.515730\pi\)
\(43\)	−73052.9	+	126531.i	−0.918824	+	1.59145i	−0.117620	+	0.993059i	\(0.537526\pi\)
							−0.801204	+	0.598391i	\(0.795807\pi\)
\(47\)	−22319.7	−	12886.3i	−0.214978	−	0.124118i	0.388645	−	0.921388i	\(-0.372943\pi\)
							−0.603623	+	0.797270i	\(0.706277\pi\)