Embedded newform 117.2.t.c.103.4

• Label: 117.2.t.c.103.4
• Level: $117$
• Weight: $2$
• Character: 117.103
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	\( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.4
Root			\(1.65391 + 0.514376i\) of defining polynomial
Character	\(\chi\)	\(=\)	117.103
Dual form			117.2.t.c.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.929969 + 0.536918i) q^{2} +(-0.744247 + 1.56400i) q^{3} +(-0.423439 + 0.733417i) q^{4} +(-1.10543 - 0.638222i) q^{5} +(-0.147613 - 1.85407i) q^{6} +(-0.890926 + 0.514376i) q^{7} -3.05708i q^{8} +(-1.89219 - 2.32800i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.929969	+	0.536918i	−0.657587	+	0.379658i	−0.791357	−	0.611354i	\(-0.790625\pi\)
							0.133770	+	0.991012i	\(0.457292\pi\)
\(3\)	−0.744247	+	1.56400i	−0.429691	+	0.902976i
							\(5\)	−1.10543	−	0.638222i	−0.494365	−	0.285422i	0.232019	−	0.972711i	\(-0.425467\pi\)
−0.726383	+	0.687290i	\(0.758800\pi\)
\(7\)	−0.890926	+	0.514376i	−0.336738	+	0.194416i	−0.658829	−	0.752293i	\(-0.728948\pi\)
							0.322090	+	0.946709i	\(0.395614\pi\)
\(11\)	−4.03796	+	2.33132i	−1.21749	+	0.702918i	−0.964380	−	0.264519i	\(-0.914787\pi\)
							−0.253110	+	0.967438i	\(0.581453\pi\)
\(13\)	2.29741	+	2.77883i	0.637187	+	0.770709i
							\(17\)	−0.476187			−0.115492			−0.0577461	−	0.998331i	\(-0.518391\pi\)
−0.0577461	+	0.998331i	\(0.518391\pi\)
\(19\)			6.69096i			1.53501i	0.641042	+	0.767505i	\(0.278502\pi\)
							−0.641042	+	0.767505i	\(0.721498\pi\)
\(23\)	−0.479867	+	0.831155i	−0.100059	+	0.173308i	−0.911709	−	0.410837i	\(-0.865237\pi\)
							0.811650	+	0.584145i	\(0.198570\pi\)
\(29\)	4.68880	+	8.12123i	0.870687	+	1.50807i	0.861287	+	0.508119i	\(0.169659\pi\)
							0.00940050	+	0.999956i	\(0.497008\pi\)
\(31\)	1.66927	+	0.963754i	0.299810	+	0.173095i	0.642357	−	0.766405i	\(-0.277956\pi\)
							−0.342548	+	0.939500i	\(0.611290\pi\)
\(37\)		−	4.94666i		−	0.813226i	−0.913601	−	0.406613i	\(-0.866710\pi\)
							0.913601	−	0.406613i	\(-0.133290\pi\)
\(41\)	1.31994	+	0.762068i	0.206140	+	0.119015i	0.599516	−	0.800363i	\(-0.295360\pi\)
							−0.393376	+	0.919378i	\(0.628693\pi\)
\(43\)	−1.31426	−	2.27637i	−0.200423	−	0.347143i	0.748242	−	0.663426i	\(-0.230898\pi\)
							−0.948665	+	0.316283i	\(0.897565\pi\)
\(47\)	−5.92316	+	3.41974i	−0.863982	+	0.498820i	−0.865344	−	0.501179i	\(-0.832900\pi\)
							0.00136148	+	0.999999i	\(0.499567\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.t.c.103.4	yes	20
3.2	odd	2		351.2.t.c.64.7		20
9.2	odd	6		351.2.t.c.181.4		20
9.4	even	3		1053.2.b.j.649.4		10
9.5	odd	6		1053.2.b.i.649.7		10
9.7	even	3	inner	117.2.t.c.25.7	yes	20
13.12	even	2	inner	117.2.t.c.103.7	yes	20
39.38	odd	2		351.2.t.c.64.4		20
117.25	even	6	inner	117.2.t.c.25.4	✓	20
117.38	odd	6		351.2.t.c.181.7		20
117.77	odd	6		1053.2.b.i.649.4		10
117.103	even	6		1053.2.b.j.649.7		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.4	✓	20	117.25	even	6	inner
117.2.t.c.25.7	yes	20	9.7	even	3	inner
117.2.t.c.103.4	yes	20	1.1	even	1	trivial
117.2.t.c.103.7	yes	20	13.12	even	2	inner
351.2.t.c.64.4		20	39.38	odd	2
351.2.t.c.64.7		20	3.2	odd	2
351.2.t.c.181.4		20	9.2	odd	6
351.2.t.c.181.7		20	117.38	odd	6
1053.2.b.i.649.4		10	117.77	odd	6
1053.2.b.i.649.7		10	9.5	odd	6
1053.2.b.j.649.4		10	9.4	even	3
1053.2.b.j.649.7		10	117.103	even	6