Embedded newform 18.7.d.a.5.5

• Label: 18.7.d.a.5.5
• 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			5.5
Root			\(8.88570i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.5
Dual form			18.7.d.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.89898 - 2.82843i) q^{2} +(-14.9408 + 22.4894i) q^{3} +(16.0000 - 27.7128i) q^{4} +(202.253 + 116.771i) q^{5} +(-9.58533 + 152.434i) q^{6} +(95.5752 + 165.541i) q^{7} -181.019i q^{8} +(-282.543 - 672.020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10}) \)

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
							\(3\)	−14.9408	+	22.4894i	−0.553364	+	0.832939i
\(5\)	202.253	+	116.771i	1.61802	+	0.934165i								0.987431	+	0.158050i	\(0.0505206\pi\)
							0.630591	+	0.776116i	\(0.282813\pi\)
\(7\)	95.5752	+	165.541i	0.278645	+	0.482627i	0.971048	−	0.238884i	\(-0.0767814\pi\)
							−0.692403	+	0.721511i	\(0.743448\pi\)
\(11\)	−673.077	+	388.601i	−0.505693	+	0.291962i	−0.731061	−	0.682312i	\(-0.760975\pi\)
							0.225369	+	0.974274i	\(0.427641\pi\)
\(13\)	45.5802	−	78.9472i	0.0207466	−	0.0359341i	−0.855466	−	0.517859i	\(-0.826729\pi\)
							0.876212	+	0.481925i	\(0.160062\pi\)
\(17\)		−	7047.39i		−	1.43444i	−0.696848	−	0.717219i	\(-0.745415\pi\)
							0.696848	−	0.717219i	\(-0.254585\pi\)
\(19\)	2731.10			0.398177			0.199088	−	0.979982i	\(-0.436202\pi\)
							0.199088	+	0.979982i	\(0.436202\pi\)
\(23\)	−17228.9	−	9947.14i	−1.41604	−	0.817550i	−0.420091	−	0.907482i	\(-0.638002\pi\)
							−0.995948	+	0.0899317i	\(0.971335\pi\)
\(29\)	27104.3	−	15648.7i	1.11133	−	0.641629i	0.172159	−	0.985069i	\(-0.444926\pi\)
							0.939175	+	0.343440i	\(0.111592\pi\)
\(31\)	6174.50	−	10694.6i	0.207261	−	0.358986i	−0.743590	−	0.668636i	\(-0.766879\pi\)
							0.950851	+	0.309650i	\(0.100212\pi\)
\(37\)	−27972.0			−0.552228			−0.276114	−	0.961125i	\(-0.589047\pi\)
							−0.276114	+	0.961125i	\(0.589047\pi\)
\(41\)	−37428.2	−	21609.2i	−0.543059	−	0.313535i	0.203259	−	0.979125i	\(-0.434847\pi\)
							−0.746318	+	0.665590i	\(0.768180\pi\)
\(43\)	19256.1	+	33352.5i	0.242193	+	0.419491i	0.961339	−	0.275369i	\(-0.0887999\pi\)
							−0.719146	+	0.694860i	\(0.755467\pi\)
\(47\)	−143771.	+	83006.2i	−1.38477	+	0.799497i	−0.992720	−	0.120447i	\(-0.961567\pi\)
							−0.392050	+	0.919944i	\(0.628234\pi\)