Embedded newform 69.2.g.a.53.2

• Label: 69.2.g.a.53.2
• Level: $69$
• Weight: $2$
• Character: 69.53
• 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			53.2
Character	\(\chi\)	\(=\)	69.53
Dual form			69.2.g.a.56.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.924731 - 0.801284i) q^{2} +(-1.53710 + 0.798324i) q^{3} +(-0.0715582 - 0.497698i) q^{4} +(-1.44790 - 3.17045i) q^{5} +(2.06109 + 0.493419i) q^{6} +(-0.466598 - 1.58909i) q^{7} +(-1.65568 + 2.57628i) q^{8} +(1.72536 - 2.45421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.924731	−	0.801284i	−0.653884	−	0.566593i	0.263470	−	0.964668i	\(-0.415133\pi\)
							−0.917353	+	0.398074i	\(0.869679\pi\)
\(3\)	−1.53710	+	0.798324i	−0.887445	+	0.460913i
							\(5\)	−1.44790	−	3.17045i	−0.647519	−	1.41787i	−0.893709	−	0.448648i	\(-0.851906\pi\)
0.246189	−	0.969222i	\(-0.420821\pi\)
\(7\)	−0.466598	−	1.58909i	−0.176358	−	0.600619i	−0.999464	−	0.0327318i	\(-0.989579\pi\)
							0.823107	−	0.567887i	\(-0.192239\pi\)
\(11\)	2.29894	+	2.65311i	0.693155	+	0.799944i	0.987811	−	0.155660i	\(-0.0497504\pi\)
							−0.294656	+	0.955604i	\(0.595205\pi\)
\(13\)	−1.83424	−	0.538580i	−0.508726	−	0.149375i	0.0172895	−	0.999851i	\(-0.494496\pi\)
							−0.526015	+	0.850475i	\(0.676314\pi\)
\(17\)	0.468681	−	3.25975i	0.113672	−	0.790605i	−0.850623	−	0.525775i	\(-0.823775\pi\)
							0.964295	−	0.264829i	\(-0.0853157\pi\)
\(19\)	4.45596	−	0.640671i	1.02227	−	0.146980i	0.389262	−	0.921127i	\(-0.372730\pi\)
							0.633006	+	0.774147i	\(0.281821\pi\)
\(23\)	3.15007	−	3.61622i	0.656835	−	0.754034i
							\(29\)	−2.57085	−	0.369633i	−0.477395	−	0.0686391i	−0.100583	−	0.994929i	\(-0.532071\pi\)
−0.376812	+	0.926290i	\(0.622980\pi\)
\(31\)	−6.47426	−	4.16075i	−1.16281	−	0.747293i	−0.190662	−	0.981656i	\(-0.561063\pi\)
							−0.972149	+	0.234363i	\(0.924700\pi\)
\(37\)	−0.285933	−	0.130581i	−0.0470071	−	0.0214674i	0.391773	−	0.920062i	\(-0.371862\pi\)
							−0.438780	+	0.898595i	\(0.644589\pi\)
\(41\)	10.0644	−	4.59624i	1.57179	−	0.717812i	0.576715	−	0.816946i	\(-0.304334\pi\)
							0.995074	+	0.0991336i	\(0.0316071\pi\)
\(43\)	2.84748	+	4.43076i	0.434237	+	0.675685i	0.987553	−	0.157286i	\(-0.0502744\pi\)
							−0.553317	+	0.832971i	\(0.686638\pi\)
\(47\)		−	0.189364i		−	0.0276216i	−0.999905	−	0.0138108i	\(-0.995604\pi\)
							0.999905	−	0.0138108i	\(-0.00439626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.53.2	✓	60
3.2	odd	2	inner	69.2.g.a.53.5	yes	60
23.10	odd	22	inner	69.2.g.a.56.5	yes	60
69.56	even	22	inner	69.2.g.a.56.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.53.2	✓	60	1.1	even	1	trivial
69.2.g.a.53.5	yes	60	3.2	odd	2	inner
69.2.g.a.56.2	yes	60	69.56	even	22	inner
69.2.g.a.56.5	yes	60	23.10	odd	22	inner