Embedded newform 69.2.g.a.17.2

• Label: 69.2.g.a.17.2
• Level: $69$
• Weight: $2$
• Character: 69.17
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			17.2
Character	\(\chi\)	\(=\)	69.17
Dual form			69.2.g.a.65.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69911 - 0.775958i) q^{2} +(-0.953890 + 1.44572i) q^{3} +(0.975144 + 1.12538i) q^{4} +(3.49515 - 2.24620i) q^{5} +(2.74258 - 1.71625i) q^{6} +(2.15133 + 0.309315i) q^{7} +(0.268869 + 0.915684i) q^{8} +(-1.18019 - 2.75811i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{7}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69911	−	0.775958i	−1.20145	−	0.548685i	−0.288787	−	0.957393i	\(-0.593252\pi\)
							−0.912665	+	0.408708i	\(0.865979\pi\)
\(3\)	−0.953890	+	1.44572i	−0.550729	+	0.834684i
							\(5\)	3.49515	−	2.24620i	1.56308	−	1.00453i	0.581484	−	0.813558i	\(-0.302472\pi\)
0.981595	−	0.190972i	\(-0.0611640\pi\)
\(7\)	2.15133	+	0.309315i	0.813127	+	0.116910i	0.536319	−	0.844015i	\(-0.319814\pi\)
							0.276808	+	0.960925i	\(0.410723\pi\)
\(11\)	−0.121890	−	0.266901i	−0.0367511	−	0.0804737i	0.890366	−	0.455245i	\(-0.150448\pi\)
							−0.927117	+	0.374772i	\(0.877721\pi\)
\(13\)	0.183083	+	1.27337i	0.0507782	+	0.353170i	0.999332	+	0.0365531i	\(0.0116378\pi\)
							−0.948554	+	0.316617i	\(0.897453\pi\)
\(17\)	−0.579426	+	0.668694i	−0.140532	+	0.162182i	−0.821652	−	0.569989i	\(-0.806947\pi\)
							0.681121	+	0.732171i	\(0.261493\pi\)
\(19\)	−1.11743	+	0.968261i	−0.256357	+	0.222134i	−0.773552	−	0.633733i	\(-0.781522\pi\)
							0.517195	+	0.855868i	\(0.326976\pi\)
\(23\)	−3.56720	+	3.20547i	−0.743814	+	0.668387i
							\(29\)	−4.42349	−	3.83297i	−0.821421	−	0.711765i	0.139008	−	0.990291i	\(-0.455609\pi\)
−0.960429	+	0.278526i	\(0.910154\pi\)
\(31\)	−3.42481	+	1.00562i	−0.615114	+	0.180614i	−0.574424	−	0.818558i	\(-0.694774\pi\)
							−0.0406903	+	0.999172i	\(0.512956\pi\)
\(37\)	−3.43697	+	5.34804i	−0.565035	+	0.879212i	−0.999772	−	0.0213458i	\(-0.993205\pi\)
							0.434737	+	0.900557i	\(0.356841\pi\)
\(41\)	3.23715	+	5.03710i	0.505558	+	0.786663i	0.996417	−	0.0845755i	\(-0.0269534\pi\)
							−0.490859	+	0.871239i	\(0.663317\pi\)
\(43\)	−1.29634	+	4.41492i	−0.197690	+	0.673270i	0.799656	+	0.600458i	\(0.205015\pi\)
							−0.997346	+	0.0728111i	\(0.976803\pi\)
\(47\)		−	5.56179i		−	0.811270i	−0.914035	−	0.405635i	\(-0.867050\pi\)
							0.914035	−	0.405635i	\(-0.132950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.17.2	✓	60
3.2	odd	2	inner	69.2.g.a.17.5	yes	60
23.19	odd	22	inner	69.2.g.a.65.5	yes	60
69.65	even	22	inner	69.2.g.a.65.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.17.2	✓	60	1.1	even	1	trivial
69.2.g.a.17.5	yes	60	3.2	odd	2	inner
69.2.g.a.65.2	yes	60	69.65	even	22	inner
69.2.g.a.65.5	yes	60	23.19	odd	22	inner