Embedded newform 69.2.g.a.65.2

• Label: 69.2.g.a.65.2
• Level: $69$
• Weight: $2$
• Character: 69.65
• 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			65.2
Character	\(\chi\)	\(=\)	69.65
Dual form			69.2.g.a.17.2

$q$-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 \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.17.2	✓	60	69.17	even	22	inner
69.2.g.a.17.5	yes	60	23.17	odd	22	inner
69.2.g.a.65.2	yes	60	1.1	even	1	trivial
69.2.g.a.65.5	yes	60	3.2	odd	2	inner