Embedded newform 69.8.e.a.4.1

• Label: 69.8.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	69.4
Dual form			69.8.e.a.52.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.97624 - 19.6552i) q^{2} +(-25.9063 + 7.60678i) q^{3} +(-221.933 + 256.124i) q^{4} +(196.683 + 126.400i) q^{5} +(382.054 + 440.914i) q^{6} +(-227.971 - 1585.58i) q^{7} +(4372.52 + 1283.89i) q^{8} +(613.274 - 394.127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.97624	−	19.6552i	−0.793395	−	1.73729i	−0.666668	−	0.745355i	\(-0.732280\pi\)
							−0.126727	−	0.991938i	\(-0.540447\pi\)
\(3\)	−25.9063	+	7.60678i	−0.553964	+	0.162658i
							\(5\)	196.683	+	126.400i	0.703673	+	0.452223i	0.842923	−	0.538034i	\(-0.180833\pi\)
−0.139250	+	0.990257i	\(0.544469\pi\)
\(7\)	−227.971	−	1585.58i	−0.251210	−	1.74720i	−0.590971	−	0.806693i	\(-0.701255\pi\)
							0.339761	−	0.940512i	\(-0.389654\pi\)
\(11\)	2020.35	−	4423.96i	0.457670	−	1.00216i	−0.530342	−	0.847784i	\(-0.677936\pi\)
							0.988012	−	0.154375i	\(-0.0493363\pi\)
\(13\)	−493.065	+	3429.34i	−0.0622447	+	0.432921i	0.934741	+	0.355331i	\(0.115632\pi\)
							−0.996985	+	0.0775906i	\(0.975277\pi\)
\(17\)	10546.1	+	12170.8i	0.520620	+	0.600827i	0.953786	−	0.300487i	\(-0.0971492\pi\)
							−0.433166	+	0.901314i	\(0.642604\pi\)
\(19\)	9868.01	−	11388.3i	0.330059	−	0.380909i	−0.566328	−	0.824180i	\(-0.691637\pi\)
							0.896388	+	0.443271i	\(0.146182\pi\)
\(23\)	−16378.2	−	56005.2i	−0.280685	−	0.959800i
							\(29\)	−76270.1	−	88020.4i	−0.580713	−	0.670178i	0.387045	−	0.922061i	\(-0.373496\pi\)
−0.967758	+	0.251883i	\(0.918950\pi\)
\(31\)	281604.	+	82686.5i	1.69775	+	0.498504i	0.980201	−	0.198004i	\(-0.0634460\pi\)
							0.717548	+	0.696509i	\(0.245264\pi\)
\(37\)	−413579.	+	265791.i	−1.34231	+	0.862650i	−0.997117	−	0.0758811i	\(-0.975823\pi\)
							−0.345193	+	0.938532i	\(0.612187\pi\)
\(41\)	−193823.	−	124562.i	−0.439199	−	0.282256i	0.302297	−	0.953214i	\(-0.402247\pi\)
							−0.741495	+	0.670958i	\(0.765883\pi\)
\(43\)	162068.	−	47587.4i	0.310855	−	0.0912752i	−0.122584	−	0.992458i	\(-0.539118\pi\)
							0.433439	+	0.901183i	\(0.357300\pi\)
\(47\)	−1.33104e6			−1.87003			−0.935016	−	0.354605i	\(-0.884615\pi\)
							−0.935016	+	0.354605i	\(0.884615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.4.1	✓	140
23.6	even	11	inner	69.8.e.a.52.1	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.4.1	✓	140	1.1	even	1	trivial
69.8.e.a.52.1	yes	140	23.6	even	11	inner