Embedded newform 189.4.e.h.109.4

• Label: 189.4.e.h.109.4
• Level: $189$
• Weight: $4$
• Character: 189.109
• 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			109.4
Root			\(0.128480 - 0.222533i\) of defining polynomial
Character	\(\chi\)	\(=\)	189.109
Dual form			189.4.e.h.163.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.884622 - 1.53221i) q^{2} +(2.43489 - 4.21735i) q^{4} +(6.52561 + 11.3027i) q^{5} +(-18.4950 - 0.966772i) q^{7} -22.7698 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{4} - 60 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.884622	−	1.53221i	−0.312761	−	0.541718i	0.666198	−	0.745775i	\(-0.267921\pi\)
							−0.978959	+	0.204057i	\(0.934587\pi\)
\(3\)		0			0
							\(5\)	6.52561	+	11.3027i	0.583669	+	1.01094i	0.995040	+	0.0994757i	\(0.0317166\pi\)
−0.411371	+	0.911468i	\(0.634950\pi\)
\(7\)	−18.4950	−	0.966772i	−0.998637	−	0.0522008i
							\(11\)	26.4705	−	45.8483i	0.725560	−	1.25671i	−0.233183	−	0.972433i	\(-0.574914\pi\)
0.958743	−	0.284274i	\(-0.0917525\pi\)
\(13\)	−71.7001			−1.52970			−0.764848	−	0.644211i	\(-0.777186\pi\)
							−0.764848	+	0.644211i	\(0.777186\pi\)
\(17\)	53.2866	−	92.2950i	0.760229	−	1.31676i	−0.182504	−	0.983205i	\(-0.558420\pi\)
							0.942732	−	0.333550i	\(-0.108247\pi\)
\(19\)	−26.9643	−	46.7035i	−0.325580	−	0.563921i	0.656049	−	0.754718i	\(-0.272226\pi\)
							−0.981630	+	0.190797i	\(0.938893\pi\)
\(23\)	9.32943	+	16.1590i	0.0845792	+	0.146495i	0.905212	−	0.424961i	\(-0.139712\pi\)
							−0.820633	+	0.571456i	\(0.806379\pi\)
\(29\)	−261.806			−1.67642			−0.838208	−	0.545350i	\(-0.816397\pi\)
							−0.838208	+	0.545350i	\(0.816397\pi\)
\(31\)	61.1632	−	105.938i	0.354362	−	0.613773i	−0.632646	−	0.774441i	\(-0.718031\pi\)
							0.987009	+	0.160667i	\(0.0513646\pi\)
\(37\)	−139.069	−	240.874i	−0.617912	−	1.07026i	−0.989866	−	0.142004i	\(-0.954645\pi\)
							0.371954	−	0.928251i	\(-0.378688\pi\)
\(41\)	31.3788			0.119525			0.0597627	−	0.998213i	\(-0.480966\pi\)
							0.0597627	+	0.998213i	\(0.480966\pi\)
\(43\)	−347.239			−1.23148			−0.615739	−	0.787951i	\(-0.711142\pi\)
							−0.615739	+	0.787951i	\(0.711142\pi\)
\(47\)	271.109	+	469.575i	0.841390	+	1.45733i	0.888720	+	0.458451i	\(0.151596\pi\)
							−0.0473295	+	0.998879i	\(0.515071\pi\)