Embedded newform 18.7.d.a.5.2

• Label: 18.7.d.a.5.2
• Level: $18$
• Weight: $7$
• Character: 18.5
• 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			5.2
Root			\(7.20150i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.5
Dual form			18.7.d.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.89898 + 2.82843i) q^{2} +(-11.1983 + 24.5683i) q^{3} +(16.0000 - 27.7128i) q^{4} +(-39.5602 - 22.8401i) q^{5} +(-14.6295 - 152.033i) q^{6} +(-245.097 - 424.521i) q^{7} +181.019i q^{8} +(-478.198 - 550.243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.89898	+	2.82843i	−0.612372	+	0.353553i
							\(3\)	−11.1983	+	24.5683i	−0.414750	+	0.909935i
\(5\)	−39.5602	−	22.8401i	−0.316482	−	0.182721i								0.333341	−	0.942806i	\(-0.391824\pi\)
							−0.649823	+	0.760085i	\(0.725157\pi\)
\(7\)	−245.097	−	424.521i	−0.714570	−	1.23767i	−0.963125	−	0.269053i	\(-0.913289\pi\)
							0.248556	−	0.968618i	\(-0.420044\pi\)
\(11\)	−873.336	+	504.221i	−0.656151	+	0.378829i	−0.790809	−	0.612063i	\(-0.790340\pi\)
							0.134658	+	0.990892i	\(0.457006\pi\)
\(13\)	466.801	−	808.523i	0.212472	−	0.368012i	−0.740016	−	0.672590i	\(-0.765182\pi\)
							0.952488	+	0.304577i	\(0.0985152\pi\)
\(17\)			8090.59i			1.64677i	0.567482	+	0.823386i	\(0.307918\pi\)
							−0.567482	+	0.823386i	\(0.692082\pi\)
\(19\)	−7727.36			−1.12660			−0.563301	−	0.826252i	\(-0.690469\pi\)
							−0.563301	+	0.826252i	\(0.690469\pi\)
\(23\)	−11848.0	−	6840.45i	−0.973782	−	0.562213i	−0.0733950	−	0.997303i	\(-0.523383\pi\)
							−0.900387	+	0.435090i	\(0.856717\pi\)
\(29\)	−1964.70	+	1134.32i	−0.0805570	+	0.0465096i	−0.539737	−	0.841833i	\(-0.681476\pi\)
							0.459180	+	0.888343i	\(0.348143\pi\)
\(31\)	−17062.6	+	29553.2i	−0.572742	+	0.992019i	0.423541	+	0.905877i	\(0.360787\pi\)
							−0.996283	+	0.0861417i	\(0.972546\pi\)
\(37\)	92058.0			1.81743			0.908713	−	0.417422i	\(-0.137066\pi\)
							0.908713	+	0.417422i	\(0.137066\pi\)
\(41\)	−31021.6	−	17910.3i	−0.450103	−	0.259867i	0.257771	−	0.966206i	\(-0.417012\pi\)
							−0.707874	+	0.706339i	\(0.750345\pi\)
\(43\)	34570.9	+	59878.5i	0.434816	+	0.753123i	0.997281	−	0.0736985i	\(-0.0234803\pi\)
							−0.562465	+	0.826821i	\(0.690147\pi\)
\(47\)	13211.1	−	7627.46i	0.127247	−	0.0734659i	−0.435025	−	0.900418i	\(-0.643261\pi\)
							0.562272	+	0.826952i	\(0.309927\pi\)