Embedded newform 117.4.t.a.103.27

• Label: 117.4.t.a.103.27
• 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			103.27
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73586 - 1.00220i) q^{2} +(-3.28702 - 4.02436i) q^{3} +(-1.99119 + 3.44883i) q^{4} +(16.4185 + 9.47923i) q^{5} +(-9.73904 - 3.69149i) q^{6} +(-21.8241 + 12.6002i) q^{7} +24.0175i q^{8} +(-5.39099 + 26.4563i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.73586	−	1.00220i	0.613720	−	0.354332i	−0.160700	−	0.987003i	\(-0.551375\pi\)
							0.774420	+	0.632672i	\(0.218042\pi\)
\(3\)	−3.28702	−	4.02436i	−0.632587	−	0.774489i
							\(5\)	16.4185	+	9.47923i	1.46852	+	0.847848i	0.999378	−	0.0352764i	\(-0.0112311\pi\)
0.469139	+	0.883125i	\(0.344564\pi\)
\(7\)	−21.8241	+	12.6002i	−1.17839	+	0.680346i	−0.955642	−	0.294530i	\(-0.904837\pi\)
							−0.222751	+	0.974875i	\(0.571504\pi\)
\(11\)	35.8629	−	20.7054i	0.983005	−	0.567538i	0.0798292	−	0.996809i	\(-0.474563\pi\)
							0.903176	+	0.429270i	\(0.141229\pi\)
\(13\)	−32.0646	+	34.1886i	−0.684085	+	0.729402i
							\(17\)	126.445			1.80397			0.901984	−	0.431770i	\(-0.142111\pi\)
0.901984	+	0.431770i	\(0.142111\pi\)
\(19\)		−	10.9807i		−	0.132587i	−0.997800	−	0.0662935i	\(-0.978883\pi\)
							0.997800	−	0.0662935i	\(-0.0211174\pi\)
\(23\)	−65.9393	+	114.210i	−0.597795	+	1.03541i	0.395351	+	0.918530i	\(0.370623\pi\)
							−0.993146	+	0.116881i	\(0.962710\pi\)
\(29\)	−52.0986	−	90.2374i	−0.333602	−	0.577816i	0.649613	−	0.760265i	\(-0.274931\pi\)
							−0.983215	+	0.182449i	\(0.941598\pi\)
\(31\)	−126.795	−	73.2050i	−0.734614	−	0.424129i	0.0854940	−	0.996339i	\(-0.472753\pi\)
							−0.820108	+	0.572209i	\(0.806086\pi\)
\(37\)			48.0579i			0.213532i	0.994284	+	0.106766i	\(0.0340495\pi\)
							−0.994284	+	0.106766i	\(0.965950\pi\)
\(41\)	22.6699	+	13.0885i	0.0863523	+	0.0498555i	0.542554	−	0.840021i	\(-0.317457\pi\)
							−0.456202	+	0.889876i	\(0.650791\pi\)
\(43\)	−32.2583	−	55.8730i	−0.114403	−	0.198152i	0.803138	−	0.595793i	\(-0.203162\pi\)
							−0.917541	+	0.397641i	\(0.869829\pi\)
\(47\)	119.978	−	69.2691i	0.372352	−	0.214978i	−0.302134	−	0.953266i	\(-0.597699\pi\)
							0.674485	+	0.738288i	\(0.264366\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.27	yes	80
3.2	odd	2		351.4.t.a.64.14		80
9.2	odd	6		351.4.t.a.181.27		80
9.7	even	3	inner	117.4.t.a.25.14	✓	80
13.12	even	2	inner	117.4.t.a.103.14	yes	80
39.38	odd	2		351.4.t.a.64.27		80
117.25	even	6	inner	117.4.t.a.25.27	yes	80
117.38	odd	6		351.4.t.a.181.14		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.14	✓	80	9.7	even	3	inner
117.4.t.a.25.27	yes	80	117.25	even	6	inner
117.4.t.a.103.14	yes	80	13.12	even	2	inner
117.4.t.a.103.27	yes	80	1.1	even	1	trivial
351.4.t.a.64.14		80	3.2	odd	2
351.4.t.a.64.27		80	39.38	odd	2
351.4.t.a.181.14		80	117.38	odd	6
351.4.t.a.181.27		80	9.2	odd	6