Embedded newform 117.4.t.a.103.16

• Label: 117.4.t.a.103.16
• 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			103.16
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12540 + 0.649749i) q^{2} +(-1.28267 - 5.03535i) q^{3} +(-3.15565 + 5.46575i) q^{4} +(4.00874 + 2.31444i) q^{5} +(4.71523 + 4.83337i) q^{6} +(7.19572 - 4.15445i) q^{7} -18.5975i q^{8} +(-23.7095 + 12.9174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12540	+	0.649749i	−0.397889	+	0.229721i	−0.685573	−	0.728004i	\(-0.740448\pi\)
							0.287684	+	0.957725i	\(0.407115\pi\)
\(3\)	−1.28267	−	5.03535i	−0.246849	−	0.969054i
							\(5\)	4.00874	+	2.31444i	0.358552	+	0.207010i	0.668446	−	0.743761i	\(-0.266960\pi\)
−0.309893	+	0.950771i	\(0.600293\pi\)
\(7\)	7.19572	−	4.15445i	0.388532	−	0.224319i	−0.292992	−	0.956115i	\(-0.594651\pi\)
							0.681524	+	0.731796i	\(0.261317\pi\)
\(11\)	32.0839	−	18.5236i	0.879422	−	0.507735i	0.00895441	−	0.999960i	\(-0.497150\pi\)
							0.870468	+	0.492225i	\(0.163816\pi\)
\(13\)	−29.3490	−	36.5464i	−0.626149	−	0.779704i
							\(17\)	−98.9278			−1.41138			−0.705692	−	0.708519i	\(-0.749364\pi\)
−0.705692	+	0.708519i	\(0.749364\pi\)
\(19\)		−	138.591i		−	1.67341i	−0.547651	−	0.836707i	\(-0.684478\pi\)
							0.547651	−	0.836707i	\(-0.315522\pi\)
\(23\)	35.5622	−	61.5956i	0.322401	−	0.558416i	−0.658582	−	0.752509i	\(-0.728843\pi\)
							0.980983	+	0.194094i	\(0.0621766\pi\)
\(29\)	1.75429	+	3.03851i	0.0112332	+	0.0194565i	0.871587	−	0.490240i	\(-0.163091\pi\)
							−0.860354	+	0.509697i	\(0.829758\pi\)
\(31\)	109.569	+	63.2597i	0.634812	+	0.366509i	0.782613	−	0.622508i	\(-0.213886\pi\)
							−0.147801	+	0.989017i	\(0.547220\pi\)
\(37\)		−	256.310i		−	1.13884i	−0.822047	−	0.569420i	\(-0.807168\pi\)
							0.822047	−	0.569420i	\(-0.192832\pi\)
\(41\)	390.869	+	225.669i	1.48887	+	0.859598i	0.999919	−	0.0127157i	\(-0.00404764\pi\)
							0.488947	+	0.872313i	\(0.337381\pi\)
\(43\)	−202.997	−	351.602i	−0.719926	−	1.24695i	−0.961029	−	0.276449i	\(-0.910842\pi\)
							0.241103	−	0.970500i	\(-0.422491\pi\)
\(47\)	−498.197	+	287.634i	−1.54616	+	0.892676i	−0.547730	+	0.836655i	\(0.684508\pi\)
							−0.998430	+	0.0560205i	\(0.982159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.16	yes	80
3.2	odd	2		351.4.t.a.64.25		80
9.2	odd	6		351.4.t.a.181.16		80
9.7	even	3	inner	117.4.t.a.25.25	yes	80
13.12	even	2	inner	117.4.t.a.103.25	yes	80
39.38	odd	2		351.4.t.a.64.16		80
117.25	even	6	inner	117.4.t.a.25.16	✓	80
117.38	odd	6		351.4.t.a.181.25		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.16	✓	80	117.25	even	6	inner
117.4.t.a.25.25	yes	80	9.7	even	3	inner
117.4.t.a.103.16	yes	80	1.1	even	1	trivial
117.4.t.a.103.25	yes	80	13.12	even	2	inner
351.4.t.a.64.16		80	39.38	odd	2
351.4.t.a.64.25		80	3.2	odd	2
351.4.t.a.181.16		80	9.2	odd	6
351.4.t.a.181.25		80	117.38	odd	6