Embedded newform 18.7.d.a.11.3

• Label: 18.7.d.a.11.3
• 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			11.3
Root			\(11.8022i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.11
Dual form			18.7.d.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.89898 - 2.82843i) q^{2} +(25.6924 - 8.30052i) q^{3} +(16.0000 + 27.7128i) q^{4} +(-9.39126 + 5.42205i) q^{5} +(-149.344 - 32.0051i) q^{6} +(322.041 - 557.792i) q^{7} -181.019i q^{8} +(591.203 - 426.521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10}) \)

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
							\(3\)	25.6924	−	8.30052i	0.951572	−	0.307427i
\(5\)	−9.39126	+	5.42205i	−0.0751301	+	0.0433764i								−0.537094	−	0.843522i	\(-0.680478\pi\)
							0.461964	+	0.886898i	\(0.347145\pi\)
\(7\)	322.041	−	557.792i	0.938896	−	1.62622i	0.171360	−	0.985208i	\(-0.445184\pi\)
							0.767535	−	0.641007i	\(-0.221483\pi\)
\(11\)	−1058.59	−	611.175i	−0.795332	−	0.459185i	0.0465046	−	0.998918i	\(-0.485192\pi\)
							−0.841836	+	0.539733i	\(0.818525\pi\)
\(13\)	1163.39	+	2015.05i	0.529535	+	0.917181i	0.999407	+	0.0344462i	\(0.0109667\pi\)
							−0.469872	+	0.882735i	\(0.655700\pi\)
\(17\)			3382.18i			0.688415i	0.938894	+	0.344207i	\(0.111852\pi\)
							−0.938894	+	0.344207i	\(0.888148\pi\)
\(19\)	6234.31			0.908924			0.454462	−	0.890766i	\(-0.349831\pi\)
							0.454462	+	0.890766i	\(0.349831\pi\)
\(23\)	−12444.3	+	7184.71i	−1.02279	+	0.590508i	−0.914911	−	0.403656i	\(-0.867739\pi\)
							−0.107879	+	0.994164i	\(0.534406\pi\)
\(29\)	11757.8	+	6788.37i	0.482095	+	0.278338i	0.721289	−	0.692634i	\(-0.243550\pi\)
							−0.239194	+	0.970972i	\(0.576883\pi\)
\(31\)	−4393.16	−	7609.18i	−0.147466	−	0.255419i	0.782824	−	0.622243i	\(-0.213778\pi\)
							−0.930290	+	0.366824i	\(0.880445\pi\)
\(37\)	−27898.6			−0.550779			−0.275389	−	0.961333i	\(-0.588807\pi\)
							−0.275389	+	0.961333i	\(0.588807\pi\)
\(41\)	−47968.4	+	27694.6i	−0.695991	+	0.401831i	−0.805852	−	0.592116i	\(-0.798293\pi\)
							0.109862	+	0.993947i	\(0.464959\pi\)
\(43\)	−26959.0	+	46694.4i	−0.339077	+	0.587299i	−0.984259	−	0.176730i	\(-0.943448\pi\)
							0.645182	+	0.764029i	\(0.276781\pi\)
\(47\)	152425.	+	88002.4i	1.46812	+	0.847620i	0.999362	−	0.0357064i	\(-0.0113681\pi\)
							0.468758	+	0.883326i	\(0.344701\pi\)