Embedded newform 117.4.t.a.103.39

• Label: 117.4.t.a.103.39
• 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			103.39
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.51411 - 2.60622i) q^{2} +(-3.21970 + 4.07843i) q^{3} +(9.58480 - 16.6014i) q^{4} +(-14.3282 - 8.27241i) q^{5} +(-3.90478 + 26.8017i) q^{6} +(26.6627 - 15.3937i) q^{7} -58.2209i q^{8} +(-6.26710 - 26.2626i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.51411	−	2.60622i	1.59598	−	0.921439i	0.603728	−	0.797190i	\(-0.293681\pi\)
							0.992251	−	0.124249i	\(-0.0396520\pi\)
\(3\)	−3.21970	+	4.07843i	−0.619631	+	0.784893i
							\(5\)	−14.3282	−	8.27241i	−1.28156	−	0.739907i	−0.304423	−	0.952537i	\(-0.598464\pi\)
−0.977133	+	0.212630i	\(0.931797\pi\)
\(7\)	26.6627	−	15.3937i	1.43965	−	0.831184i	0.441828	−	0.897100i	\(-0.354330\pi\)
							0.997825	+	0.0659158i	\(0.0209969\pi\)
\(11\)	20.0591	−	11.5811i	0.549823	−	0.317440i	−0.199228	−	0.979953i	\(-0.563843\pi\)
							0.749051	+	0.662513i	\(0.230510\pi\)
\(13\)	−30.6700	+	35.4450i	−0.654334	+	0.756206i
							\(17\)	−6.16225			−0.0879155			−0.0439578	−	0.999033i	\(-0.513997\pi\)
−0.0439578	+	0.999033i	\(0.513997\pi\)
\(19\)			50.9595i			0.615311i	0.951498	+	0.307655i	\(0.0995444\pi\)
							−0.951498	+	0.307655i	\(0.900456\pi\)
\(23\)	47.7300	−	82.6707i	0.432712	−	0.749480i	−0.564393	−	0.825506i	\(-0.690890\pi\)
							0.997106	+	0.0760260i	\(0.0242232\pi\)
\(29\)	67.8473	+	117.515i	0.434446	+	0.752482i	0.997250	−	0.0741079i	\(-0.0236109\pi\)
							−0.562804	+	0.826590i	\(0.690278\pi\)
\(31\)	256.578	+	148.136i	1.48654	+	0.858256i	0.999882	−	0.0153345i	\(-0.00488131\pi\)
							0.486661	+	0.873591i	\(0.338215\pi\)
\(37\)			263.986i			1.17294i	0.809969	+	0.586472i	\(0.199484\pi\)
							−0.809969	+	0.586472i	\(0.800516\pi\)
\(41\)	−264.258	−	152.570i	−1.00659	−	0.581155i	−0.0963990	−	0.995343i	\(-0.530732\pi\)
							−0.910192	+	0.414187i	\(0.864066\pi\)
\(43\)	−75.4134	−	130.620i	−0.267452	−	0.463241i	0.700751	−	0.713406i	\(-0.252848\pi\)
							−0.968203	+	0.250165i	\(0.919515\pi\)
\(47\)	122.276	−	70.5961i	0.379485	−	0.219096i	−0.298109	−	0.954532i	\(-0.596356\pi\)
							0.677594	+	0.735436i	\(0.263023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.39	yes	80
3.2	odd	2		351.4.t.a.64.2		80
9.2	odd	6		351.4.t.a.181.39		80
9.7	even	3	inner	117.4.t.a.25.2	✓	80
13.12	even	2	inner	117.4.t.a.103.2	yes	80
39.38	odd	2		351.4.t.a.64.39		80
117.25	even	6	inner	117.4.t.a.25.39	yes	80
117.38	odd	6		351.4.t.a.181.2		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.2	✓	80	9.7	even	3	inner
117.4.t.a.25.39	yes	80	117.25	even	6	inner
117.4.t.a.103.2	yes	80	13.12	even	2	inner
117.4.t.a.103.39	yes	80	1.1	even	1	trivial
351.4.t.a.64.2		80	3.2	odd	2
351.4.t.a.64.39		80	39.38	odd	2
351.4.t.a.181.2		80	117.38	odd	6
351.4.t.a.181.39		80	9.2	odd	6