Embedded newform 69.8.e.a.4.3

• Label: 69.8.e.a.4.3
• Level: $69$
• Weight: $8$
• Character: 69.4
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(14\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	69.4
Dual form			69.8.e.a.52.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.20815 - 15.7837i) q^{2} +(-25.9063 + 7.60678i) q^{3} +(-113.344 + 130.806i) q^{4} +(267.037 + 171.614i) q^{5} +(306.799 + 354.065i) q^{6} +(122.562 + 852.436i) q^{7} +(750.547 + 220.380i) q^{8} +(613.274 - 394.127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{2}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.20815	−	15.7837i	−0.637117	−	1.39509i	−0.902391	−	0.430918i	\(-0.858190\pi\)
							0.265275	−	0.964173i	\(-0.414537\pi\)
\(3\)	−25.9063	+	7.60678i	−0.553964	+	0.162658i
							\(5\)	267.037	+	171.614i	0.955381	+	0.613986i	0.922716	−	0.385480i	\(-0.125964\pi\)
0.0326648	+	0.999466i	\(0.489601\pi\)
\(7\)	122.562	+	852.436i	0.135055	+	0.939331i	0.938826	+	0.344392i	\(0.111915\pi\)
							−0.803771	+	0.594939i	\(0.797176\pi\)
\(11\)	−2060.67	+	4512.24i	−0.466804	+	1.02216i	0.519080	+	0.854726i	\(0.326275\pi\)
							−0.985884	+	0.167432i	\(0.946453\pi\)
\(13\)	1782.31	−	12396.3i	0.225000	−	1.56491i	−0.493727	−	0.869617i	\(-0.664366\pi\)
							0.718727	−	0.695292i	\(-0.244725\pi\)
\(17\)	−20933.2	−	24158.2i	−1.03339	−	1.19259i	−0.981008	−	0.193967i	\(-0.937865\pi\)
							−0.0523806	−	0.998627i	\(-0.516681\pi\)
\(19\)	−6215.51	+	7173.09i	−0.207893	+	0.239921i	−0.850115	−	0.526598i	\(-0.823467\pi\)
							0.642222	+	0.766519i	\(0.278013\pi\)
\(23\)	−40579.8	−	41929.7i	−0.695445	−	0.718579i
							\(29\)	77894.7	+	89895.2i	0.593082	+	0.684453i	0.970365	−	0.241645i	\(-0.0776869\pi\)
−0.377283	+	0.926098i	\(0.623141\pi\)
\(31\)	63872.7	+	18754.7i	0.385079	+	0.113069i	0.468541	−	0.883442i	\(-0.344780\pi\)
							−0.0834626	+	0.996511i	\(0.526598\pi\)
\(37\)	215503.	−	138495.i	0.699435	−	0.449500i	−0.141994	−	0.989868i	\(-0.545351\pi\)
							0.841429	+	0.540368i	\(0.181715\pi\)
\(41\)	−319448.	−	205297.i	−0.723863	−	0.465199i	0.126116	−	0.992016i	\(-0.459749\pi\)
							−0.849979	+	0.526817i	\(0.823385\pi\)
\(43\)	−763098.	+	224066.i	−1.46366	+	0.429770i	−0.914033	−	0.405640i	\(-0.867049\pi\)
							−0.549628	+	0.835409i	\(0.685231\pi\)
\(47\)	−1.13488e6			−1.59444			−0.797220	−	0.603689i	\(-0.793697\pi\)
							−0.797220	+	0.603689i	\(0.793697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.4.3	✓	140
23.6	even	11	inner	69.8.e.a.52.3	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.4.3	✓	140	1.1	even	1	trivial
69.8.e.a.52.3	yes	140	23.6	even	11	inner