Embedded newform 117.2.l.b.88.5

• Label: 117.2.l.b.88.5
• Level: $117$
• Weight: $2$
• Character: 117.88
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			88.5
Character	\(\chi\)	\(=\)	117.88
Dual form			117.2.l.b.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.571953i q^{2} +(-0.656000 - 1.60302i) q^{3} +1.67287 q^{4} +(0.796103 - 0.459630i) q^{5} +(-0.916851 + 0.375201i) q^{6} +(-1.67386 + 0.966405i) q^{7} -2.10071i q^{8} +(-2.13933 + 2.10316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - q^{3} - 20 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} + 7 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	0.571953i		−	0.404432i	−0.979341	−	0.202216i	\(-0.935186\pi\)
							0.979341	−	0.202216i	\(-0.0648143\pi\)
\(3\)	−0.656000	−	1.60302i	−0.378742	−	0.925502i
							\(5\)	0.796103	−	0.459630i	0.356028	−	0.205553i	−0.311309	−	0.950309i	\(-0.600767\pi\)
0.667337	+	0.744756i	\(0.267434\pi\)
\(7\)	−1.67386	+	0.966405i	−0.632660	+	0.365267i	−0.781782	−	0.623552i	\(-0.785689\pi\)
							0.149121	+	0.988819i	\(0.452356\pi\)
\(11\)		−	4.20402i		−	1.26756i	−0.773514	−	0.633780i	\(-0.781503\pi\)
							0.773514	−	0.633780i	\(-0.218497\pi\)
\(13\)	2.21656	+	2.84374i	0.614762	+	0.788713i
							\(17\)	−1.20321	+	2.08402i	−0.291822	+	0.505450i	−0.974241	−	0.225511i	\(-0.927595\pi\)
0.682419	+	0.730961i	\(0.260928\pi\)
\(19\)	1.60854	+	0.928692i	0.369025	+	0.213056i	0.673032	−	0.739613i	\(-0.264991\pi\)
							−0.304008	+	0.952670i	\(0.598325\pi\)
\(23\)	−4.11198	+	7.12216i	−0.857407	+	1.48507i	0.0169865	+	0.999856i	\(0.494593\pi\)
							−0.874394	+	0.485217i	\(0.838741\pi\)
\(29\)	4.73399			0.879080			0.439540	−	0.898223i	\(-0.355142\pi\)
							0.439540	+	0.898223i	\(0.355142\pi\)
\(31\)	4.29178	−	2.47786i	0.770826	−	0.445037i	−0.0623429	−	0.998055i	\(-0.519857\pi\)
							0.833169	+	0.553018i	\(0.186524\pi\)
\(37\)	−0.959703	+	0.554085i	−0.157774	+	0.0910910i	−0.576808	−	0.816879i	\(-0.695702\pi\)
							0.419034	+	0.907970i	\(0.362369\pi\)
\(41\)	−0.490992	−	0.283475i	−0.0766801	−	0.0442713i	0.461170	−	0.887312i	\(-0.347430\pi\)
							−0.537850	+	0.843041i	\(0.680763\pi\)
\(43\)	−4.79362	−	8.30280i	−0.731021	−	1.26616i	−0.956447	−	0.291905i	\(-0.905711\pi\)
							0.225427	−	0.974260i	\(-0.427622\pi\)
\(47\)	1.35155	+	0.780318i	0.197144	+	0.113821i	0.595323	−	0.803487i	\(-0.297024\pi\)
							−0.398179	+	0.917308i	\(0.630358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.l.b.88.5	yes	22
3.2	odd	2		351.2.l.b.127.7		22
9.4	even	3		117.2.r.b.49.7	yes	22
9.5	odd	6		351.2.r.b.10.5		22
13.4	even	6		117.2.r.b.43.7	yes	22
39.17	odd	6		351.2.r.b.316.5		22
117.4	even	6	inner	117.2.l.b.4.7	✓	22
117.95	odd	6		351.2.l.b.199.5		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.l.b.4.7	✓	22	117.4	even	6	inner
117.2.l.b.88.5	yes	22	1.1	even	1	trivial
117.2.r.b.43.7	yes	22	13.4	even	6
117.2.r.b.49.7	yes	22	9.4	even	3
351.2.l.b.127.7		22	3.2	odd	2
351.2.l.b.199.5		22	117.95	odd	6
351.2.r.b.10.5		22	9.5	odd	6
351.2.r.b.316.5		22	39.17	odd	6