Embedded newform 117.4.r.a.43.10

• Label: 117.4.r.a.43.10
• Level: $117$
• Weight: $4$
• Character: 117.43
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.90322347067\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.10
Character	\(\chi\)	\(=\)	117.43
Dual form			117.4.r.a.49.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.17897 + 1.83538i) q^{2} +(4.08181 + 3.21540i) q^{3} +(2.73723 - 4.74102i) q^{4} +(-15.6805 + 9.05312i) q^{5} +(-18.8774 - 2.72998i) q^{6} -31.1968i q^{7} -9.27065i q^{8} +(6.32242 + 26.2493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 3 q^{2} - q^{3} + 153 q^{4} - 3 q^{6} - 17 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.17897	+	1.83538i	−1.12394	+	0.648904i	−0.942403	−	0.334481i	\(-0.891439\pi\)
							−0.181533	+	0.983385i	\(0.558106\pi\)
\(3\)	4.08181	+	3.21540i	0.785546	+	0.618804i
							\(5\)	−15.6805	+	9.05312i	−1.40250	+	0.809736i	−0.994649	−	0.103311i	\(-0.967056\pi\)
−0.407854	+	0.913047i	\(0.633723\pi\)
\(7\)		−	31.1968i		−	1.68447i	−0.539111	−	0.842235i	\(-0.681240\pi\)
							0.539111	−	0.842235i	\(-0.318760\pi\)
\(11\)	4.59003	−	2.65005i	0.125813	−	0.0726383i	−0.435773	−	0.900057i	\(-0.643525\pi\)
							0.561586	+	0.827419i	\(0.310191\pi\)
\(13\)	−38.4823	−	26.7603i	−0.821005	−	0.570921i
							\(17\)	−1.95690	−	3.38945i	−0.0279187	−	0.0483566i	0.851728	−	0.523983i	\(-0.175555\pi\)
−0.879647	+	0.475627i	\(0.842221\pi\)
\(19\)	19.5736	−	11.3008i	0.236342	−	0.136452i	−0.377152	−	0.926151i	\(-0.623097\pi\)
							0.613494	+	0.789699i	\(0.289763\pi\)
\(23\)	55.9329			0.507079			0.253539	−	0.967325i	\(-0.418405\pi\)
							0.253539	+	0.967325i	\(0.418405\pi\)
\(29\)	−147.253	−	255.050i	−0.942905	−	1.63316i	−0.759892	−	0.650050i	\(-0.774748\pi\)
							−0.183014	−	0.983110i	\(-0.558585\pi\)
\(31\)	150.573	−	86.9333i	0.872377	−	0.503667i	0.00423943	−	0.999991i	\(-0.498651\pi\)
							0.868137	+	0.496324i	\(0.165317\pi\)
\(37\)	−328.880	−	189.879i	−1.46129	−	0.843674i	−0.462215	−	0.886768i	\(-0.652945\pi\)
							−0.999071	+	0.0430940i	\(0.986279\pi\)
\(41\)			132.647i			0.505268i	0.967562	+	0.252634i	\(0.0812968\pi\)
							−0.967562	+	0.252634i	\(0.918703\pi\)
\(43\)	−60.6002			−0.214917			−0.107459	−	0.994210i	\(-0.534271\pi\)
							−0.107459	+	0.994210i	\(0.534271\pi\)
\(47\)	−229.733	−	132.636i	−0.712978	−	0.411638i	0.0991849	−	0.995069i	\(-0.468376\pi\)
							−0.812163	+	0.583431i	\(0.801710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.r.a.43.10	yes	80
3.2	odd	2		351.4.r.a.316.31		80
9.4	even	3		117.4.l.a.4.10	✓	80
9.5	odd	6		351.4.l.a.199.31		80
13.10	even	6		117.4.l.a.88.31	yes	80
39.23	odd	6		351.4.l.a.127.10		80
117.23	odd	6		351.4.r.a.10.31		80
117.49	even	6	inner	117.4.r.a.49.10	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.l.a.4.10	✓	80	9.4	even	3
117.4.l.a.88.31	yes	80	13.10	even	6
117.4.r.a.43.10	yes	80	1.1	even	1	trivial
117.4.r.a.49.10	yes	80	117.49	even	6	inner
351.4.l.a.127.10		80	39.23	odd	6
351.4.l.a.199.31		80	9.5	odd	6
351.4.r.a.10.31		80	117.23	odd	6
351.4.r.a.316.31		80	3.2	odd	2