Embedded newform 117.4.t.a.103.18

• Label: 117.4.t.a.103.18
• 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			103.18
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.916837 + 0.529336i) q^{2} +(3.64187 - 3.70632i) q^{3} +(-3.43961 + 5.95757i) q^{4} +(-2.58576 - 1.49289i) q^{5} +(-1.37711 + 5.32586i) q^{6} +(-19.0506 + 10.9989i) q^{7} -15.7522i q^{8} +(-0.473631 - 26.9958i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.916837	+	0.529336i	−0.324151	+	0.187149i	−0.653241	−	0.757150i	\(-0.726591\pi\)
							0.329090	+	0.944298i	\(0.393258\pi\)
\(3\)	3.64187	−	3.70632i	0.700877	−	0.713282i
							\(5\)	−2.58576	−	1.49289i	−0.231277	−	0.133528i	0.379884	−	0.925034i	\(-0.375964\pi\)
−0.611161	+	0.791506i	\(0.709297\pi\)
\(7\)	−19.0506	+	10.9989i	−1.02864	+	0.593883i	−0.916594	−	0.399820i	\(-0.869073\pi\)
							−0.112042	+	0.993703i	\(0.535739\pi\)
\(11\)	−29.0235	+	16.7567i	−0.795538	+	0.459304i	−0.841908	−	0.539620i	\(-0.818568\pi\)
							0.0463707	+	0.998924i	\(0.485234\pi\)
\(13\)	−44.6088	−	14.3893i	−0.951713	−	0.306990i
							\(17\)	25.4910			0.363675			0.181838	−	0.983329i	\(-0.441795\pi\)
0.181838	+	0.983329i	\(0.441795\pi\)
\(19\)			74.3878i			0.898196i	0.893482	+	0.449098i	\(0.148255\pi\)
							−0.893482	+	0.449098i	\(0.851745\pi\)
\(23\)	−93.5481	+	162.030i	−0.848092	+	1.46894i	0.0348159	+	0.999394i	\(0.488916\pi\)
							−0.882908	+	0.469545i	\(0.844418\pi\)
\(29\)	−12.5886	−	21.8040i	−0.0806082	−	0.139617i	0.822903	−	0.568182i	\(-0.192353\pi\)
							−0.903511	+	0.428564i	\(0.859020\pi\)
\(31\)	−154.009	−	88.9174i	−0.892287	−	0.515162i	−0.0175974	−	0.999845i	\(-0.505602\pi\)
							−0.874690	+	0.484683i	\(0.838935\pi\)
\(37\)		−	252.938i		−	1.12386i	−0.827186	−	0.561928i	\(-0.810060\pi\)
							0.827186	−	0.561928i	\(-0.189940\pi\)
\(41\)	266.236	+	153.711i	1.01412	+	0.585504i	0.912396	−	0.409308i	\(-0.134230\pi\)
							0.101727	+	0.994812i	\(0.467563\pi\)
\(43\)	209.184	+	362.317i	0.741865	+	1.28495i	0.951645	+	0.307200i	\(0.0993920\pi\)
							−0.209780	+	0.977749i	\(0.567275\pi\)
\(47\)	330.941	−	191.069i	1.02708	−	0.592983i	0.110931	−	0.993828i	\(-0.464617\pi\)
							0.916146	+	0.400845i	\(0.131283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.18	yes	80
3.2	odd	2		351.4.t.a.64.23		80
9.2	odd	6		351.4.t.a.181.18		80
9.7	even	3	inner	117.4.t.a.25.23	yes	80
13.12	even	2	inner	117.4.t.a.103.23	yes	80
39.38	odd	2		351.4.t.a.64.18		80
117.25	even	6	inner	117.4.t.a.25.18	✓	80
117.38	odd	6		351.4.t.a.181.23		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.18	✓	80	117.25	even	6	inner
117.4.t.a.25.23	yes	80	9.7	even	3	inner
117.4.t.a.103.18	yes	80	1.1	even	1	trivial
117.4.t.a.103.23	yes	80	13.12	even	2	inner
351.4.t.a.64.18		80	39.38	odd	2
351.4.t.a.64.23		80	3.2	odd	2
351.4.t.a.181.18		80	9.2	odd	6
351.4.t.a.181.23		80	117.38	odd	6