Embedded newform 117.4.t.a.103.22

• Label: 117.4.t.a.103.22
• Level: $117$
• Weight: $4$
• Character: 117.103
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.t (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			103.22
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.457077 - 0.263894i) q^{2} +(0.870272 + 5.12276i) q^{3} +(-3.86072 + 6.68696i) q^{4} +(-14.6710 - 8.47032i) q^{5} +(1.74964 + 2.11184i) q^{6} +(18.0982 - 10.4490i) q^{7} +8.29758i q^{8} +(-25.4853 + 8.91639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.457077	−	0.263894i	0.161601	−	0.0933005i	−0.417018	−	0.908898i	\(-0.636925\pi\)
							0.578620	+	0.815598i	\(0.303592\pi\)
\(3\)	0.870272	+	5.12276i	0.167484	+	0.985875i
							\(5\)	−14.6710	−	8.47032i	−1.31222	−	0.757608i	−0.329753	−	0.944067i	\(-0.606966\pi\)
−0.982463	+	0.186459i	\(0.940299\pi\)
\(7\)	18.0982	−	10.4490i	0.977209	−	0.564192i	0.0757827	−	0.997124i	\(-0.475854\pi\)
							0.901426	+	0.432932i	\(0.142521\pi\)
\(11\)	−37.8111	+	21.8302i	−1.03641	+	0.598369i	−0.918813	−	0.394692i	\(-0.870851\pi\)
							−0.117593	+	0.993062i	\(0.537518\pi\)
\(13\)	3.29125	−	46.7565i	0.0702175	−	0.997532i
							\(17\)	−84.5583			−1.20638			−0.603188	−	0.797599i	\(-0.706103\pi\)
−0.603188	+	0.797599i	\(0.706103\pi\)
\(19\)		−	48.1226i		−	0.581057i	−0.956866	−	0.290528i	\(-0.906169\pi\)
							0.956866	−	0.290528i	\(-0.0938311\pi\)
\(23\)	−38.9639	+	67.4875i	−0.353241	+	0.611831i	−0.986815	−	0.161851i	\(-0.948254\pi\)
							0.633574	+	0.773682i	\(0.281587\pi\)
\(29\)	105.576	+	182.864i	0.676036	+	1.17093i	0.976165	+	0.217030i	\(0.0696370\pi\)
							−0.300129	+	0.953899i	\(0.597030\pi\)
\(31\)	−144.571	−	83.4679i	−0.837602	−	0.483590i	0.0188463	−	0.999822i	\(-0.494001\pi\)
							−0.856448	+	0.516233i	\(0.827334\pi\)
\(37\)			379.020i			1.68407i	0.539426	+	0.842033i	\(0.318641\pi\)
							−0.539426	+	0.842033i	\(0.681359\pi\)
\(41\)	175.824	+	101.512i	0.669733	+	0.386671i	0.795975	−	0.605329i	\(-0.206958\pi\)
							−0.126242	+	0.991999i	\(0.540292\pi\)
\(43\)	189.974	+	329.045i	0.673739	+	1.16695i	0.976836	+	0.213990i	\(0.0686460\pi\)
							−0.303097	+	0.952960i	\(0.598021\pi\)
\(47\)	−306.736	+	177.094i	−0.951959	+	0.549614i	−0.893689	−	0.448687i	\(-0.851892\pi\)
							−0.0582699	+	0.998301i	\(0.518558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.22	yes	80
3.2	odd	2		351.4.t.a.64.19		80
9.2	odd	6		351.4.t.a.181.22		80
9.7	even	3	inner	117.4.t.a.25.19	✓	80
13.12	even	2	inner	117.4.t.a.103.19	yes	80
39.38	odd	2		351.4.t.a.64.22		80
117.25	even	6	inner	117.4.t.a.25.22	yes	80
117.38	odd	6		351.4.t.a.181.19		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.19	✓	80	9.7	even	3	inner
117.4.t.a.25.22	yes	80	117.25	even	6	inner
117.4.t.a.103.19	yes	80	13.12	even	2	inner
117.4.t.a.103.22	yes	80	1.1	even	1	trivial
351.4.t.a.64.19		80	3.2	odd	2
351.4.t.a.64.22		80	39.38	odd	2
351.4.t.a.181.19		80	117.38	odd	6
351.4.t.a.181.22		80	9.2	odd	6