Embedded newform 189.4.e.h.163.3

• Label: 189.4.e.h.163.3
• Level: $189$
• Weight: $4$
• Character: 189.163
• Analytic conductor: $11.151$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	189.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1513609911\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 54*x^14 - 12*x^13 + 2361*x^12 - 966*x^11 + 29570*x^10 - 65952*x^9 + 300096*x^8 - 356610*x^7 + 531858*x^6 + 80070*x^5 + 77197*x^4 - 216*x^3 + 4878*x^2 + 486*x + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{9}\cdot 7^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.3
Root			\(-0.0640761 - 0.110983i\) of defining polynomial
Character	\(\chi\)	\(=\)	189.163
Dual form			189.4.e.h.109.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64637 + 2.85159i) q^{2} +(-1.42105 - 2.46133i) q^{4} +(-10.8582 + 18.8070i) q^{5} +(1.08769 + 18.4883i) q^{7} -16.9836 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{4} - 60 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64637	+	2.85159i	−0.582079	+	1.00819i	0.413154	+	0.910661i	\(0.364427\pi\)
							−0.995233	+	0.0975288i	\(0.968906\pi\)
\(3\)		0			0
							\(5\)	−10.8582	+	18.8070i	−0.971190	+	1.68215i	−0.279217	+	0.960228i	\(0.590075\pi\)
−0.691973	+	0.721923i	\(0.743258\pi\)
\(7\)	1.08769	+	18.4883i	0.0587298	+	0.998274i
							\(11\)	20.9820	+	36.3419i	0.575119	+	0.996136i	0.996029	+	0.0890334i	\(0.0283778\pi\)
−0.420909	+	0.907103i	\(0.638289\pi\)
\(13\)	46.0329			0.982095			0.491047	−	0.871133i	\(-0.336614\pi\)
							0.491047	+	0.871133i	\(0.336614\pi\)
\(17\)	−1.00882	−	1.74732i	−0.0143926	−	0.0249287i	0.858739	−	0.512413i	\(-0.171248\pi\)
							−0.873132	+	0.487484i	\(0.837915\pi\)
\(19\)	−36.6244	+	63.4352i	−0.442221	+	0.765950i	−0.997854	−	0.0654782i	\(-0.979143\pi\)
							0.555633	+	0.831428i	\(0.312476\pi\)
\(23\)	12.0838	−	20.9298i	0.109550	−	0.189747i	−0.806038	−	0.591864i	\(-0.798392\pi\)
							0.915588	+	0.402117i	\(0.131726\pi\)
\(29\)	90.7240			0.580932			0.290466	−	0.956885i	\(-0.406190\pi\)
							0.290466	+	0.956885i	\(0.406190\pi\)
\(31\)	−26.6026	−	46.0771i	−0.154128	−	0.266958i	0.778613	−	0.627504i	\(-0.215923\pi\)
							−0.932741	+	0.360547i	\(0.882590\pi\)
\(37\)	−33.9309	+	58.7701i	−0.150762	+	0.261128i	−0.931508	−	0.363721i	\(-0.881506\pi\)
							0.780745	+	0.624849i	\(0.214840\pi\)
\(41\)	341.211			1.29971			0.649856	−	0.760057i	\(-0.274829\pi\)
							0.649856	+	0.760057i	\(0.274829\pi\)
\(43\)	509.713			1.80769			0.903843	−	0.427863i	\(-0.140734\pi\)
							0.903843	+	0.427863i	\(0.140734\pi\)
\(47\)	−19.2351	+	33.3162i	−0.0596963	+	0.103397i	−0.894329	−	0.447410i	\(-0.852347\pi\)
							0.834633	+	0.550807i	\(0.185680\pi\)