Embedded newform 69.6.g.a.5.5

• Label: 69.6.g.a.5.5
• Level: $69$
• Weight: $6$
• Character: 69.5
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $380$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(380\)
Relative dimension:	\(38\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.59429 + 8.83533i) q^{2} +(15.4604 - 1.99434i) q^{3} +(-44.4125 - 28.5422i) q^{4} +(-6.70991 + 46.6684i) q^{5} +(-22.4879 + 141.771i) q^{6} +(-7.07279 + 3.23003i) q^{7} +(144.705 - 125.387i) q^{8} +(235.045 - 61.6665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.59429	+	8.83533i	−0.458609	+	1.56188i	0.328154	+	0.944624i	\(0.393574\pi\)
							−0.786763	+	0.617255i	\(0.788245\pi\)
\(3\)	15.4604	−	1.99434i	0.991782	−	0.127937i
							\(5\)	−6.70991	+	46.6684i	−0.120030	+	0.834830i	0.837488	+	0.546456i	\(0.184024\pi\)
−0.957518	+	0.288374i	\(0.906885\pi\)
\(7\)	−7.07279	+	3.23003i	−0.0545564	+	0.0249151i	−0.442506	−	0.896765i	\(-0.645911\pi\)
							0.387950	+	0.921680i	\(0.373183\pi\)
\(11\)	−631.951	+	185.558i	−1.57472	+	0.462378i	−0.948370	−	0.317167i	\(-0.897268\pi\)
							−0.626346	+	0.779546i	\(0.715450\pi\)
\(13\)	−263.870	+	577.794i	−0.433043	+	0.948231i	0.559780	+	0.828641i	\(0.310885\pi\)
							−0.992823	+	0.119591i	\(0.961842\pi\)
\(17\)	−560.572	+	360.258i	−0.470446	+	0.302337i	−0.754299	−	0.656531i	\(-0.772023\pi\)
							0.283854	+	0.958868i	\(0.408387\pi\)
\(19\)	255.687	−	397.857i	0.162489	−	0.252838i	−0.750456	−	0.660921i	\(-0.770166\pi\)
							0.912945	+	0.408082i	\(0.133802\pi\)
\(23\)	−1364.42	−	2138.86i	−0.537808	−	0.843067i
							\(29\)	2375.04	+	3695.63i	0.524415	+	0.816006i	0.997898	−	0.0647969i	\(-0.0206400\pi\)
−0.473483	+	0.880803i	\(0.657004\pi\)
\(31\)	302.762	+	349.406i	0.0565845	+	0.0653020i	0.783336	−	0.621598i	\(-0.213516\pi\)
							−0.726752	+	0.686900i	\(0.758971\pi\)
\(37\)	−4678.04	+	672.600i	−0.561771	+	0.0807705i	−0.417350	−	0.908746i	\(-0.637041\pi\)
							−0.144421	+	0.989516i	\(0.546132\pi\)
\(41\)	4829.15	+	694.328i	0.448654	+	0.0645067i	0.362939	−	0.931813i	\(-0.381773\pi\)
							0.0857151	+	0.996320i	\(0.472683\pi\)
\(43\)	−4690.31	−	4064.18i	−0.386839	−	0.335198i	0.439631	−	0.898178i	\(-0.355109\pi\)
							−0.826470	+	0.562980i	\(0.809655\pi\)
\(47\)			25715.7i			1.69806i	0.528344	+	0.849031i	\(0.322813\pi\)
							−0.528344	+	0.849031i	\(0.677187\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.5	✓	380
3.2	odd	2	inner	69.6.g.a.5.34	yes	380
23.14	odd	22	inner	69.6.g.a.14.34	yes	380
69.14	even	22	inner	69.6.g.a.14.5	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.5	✓	380	1.1	even	1	trivial
69.6.g.a.5.34	yes	380	3.2	odd	2	inner
69.6.g.a.14.5	yes	380	69.14	even	22	inner
69.6.g.a.14.34	yes	380	23.14	odd	22	inner