Embedded newform 69.6.g.a.5.10

• Label: 69.6.g.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75424 + 5.97440i) q^{2} +(-0.0984677 - 15.5881i) q^{3} +(-5.69594 - 3.66056i) q^{4} +(11.1842 - 77.7880i) q^{5} +(93.3025 + 26.7571i) q^{6} +(-8.66991 + 3.95942i) q^{7} +(-118.723 + 102.874i) q^{8} +(-242.981 + 3.06986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75424	+	5.97440i	−0.310109	+	1.05613i	0.646052	+	0.763293i	\(0.276419\pi\)
							−0.956161	+	0.292841i	\(0.905399\pi\)
\(3\)	−0.0984677	−	15.5881i	−0.00631670	−	0.999980i
							\(5\)	11.1842	−	77.7880i	0.200070	−	1.39151i	−0.604003	−	0.796982i	\(-0.706428\pi\)
0.804072	−	0.594532i	\(-0.202663\pi\)
\(7\)	−8.66991	+	3.95942i	−0.0668759	+	0.0305412i	−0.448572	−	0.893747i	\(-0.648067\pi\)
							0.381696	+	0.924288i	\(0.375340\pi\)
\(11\)	−490.982	+	144.165i	−1.22344	+	0.359236i	−0.828772	−	0.559586i	\(-0.810960\pi\)
							−0.394672	+	0.918822i	\(0.629142\pi\)
\(13\)	−121.662	+	266.402i	−0.199662	+	0.437199i	−0.982806	−	0.184641i	\(-0.940888\pi\)
							0.783144	+	0.621841i	\(0.213615\pi\)
\(17\)	−810.072	+	520.602i	−0.679832	+	0.436901i	−0.834458	−	0.551072i	\(-0.814219\pi\)
							0.154626	+	0.987973i	\(0.450583\pi\)
\(19\)	−219.039	+	340.832i	−0.139200	+	0.216599i	−0.903853	−	0.427843i	\(-0.859274\pi\)
							0.764654	+	0.644442i	\(0.222910\pi\)
\(23\)	2024.20	−	1529.36i	0.797874	−	0.602824i
							\(29\)	−2045.35	−	3182.62i	−0.451619	−	0.702732i	0.538553	−	0.842592i	\(-0.318971\pi\)
−0.990171	+	0.139860i	\(0.955335\pi\)
\(31\)	−5880.30	−	6786.23i	−1.09899	−	1.26831i	−0.960601	−	0.277930i	\(-0.910352\pi\)
							−0.138393	−	0.990377i	\(-0.544194\pi\)
\(37\)	−6922.21	+	995.263i	−0.831267	+	0.119518i	−0.544794	−	0.838570i	\(-0.683392\pi\)
							−0.286473	+	0.958088i	\(0.592483\pi\)
\(41\)	11887.2	+	1709.12i	1.10438	+	0.158786i	0.670303	−	0.742088i	\(-0.266164\pi\)
							0.434081	+	0.900874i	\(0.357073\pi\)
\(43\)	473.881	+	410.620i	0.0390839	+	0.0338664i	0.674184	−	0.738564i	\(-0.264496\pi\)
							−0.635100	+	0.772430i	\(0.719041\pi\)
\(47\)			9776.78i			0.645581i	0.946470	+	0.322791i	\(0.104621\pi\)
							−0.946470	+	0.322791i	\(0.895379\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.10	✓	380
3.2	odd	2	inner	69.6.g.a.5.29	yes	380
23.14	odd	22	inner	69.6.g.a.14.29	yes	380
69.14	even	22	inner	69.6.g.a.14.10	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.10	✓	380	1.1	even	1	trivial
69.6.g.a.5.29	yes	380	3.2	odd	2	inner
69.6.g.a.14.10	yes	380	69.14	even	22	inner
69.6.g.a.14.29	yes	380	23.14	odd	22	inner