Embedded newform 69.4.e.b.16.4

• Label: 69.4.e.b.16.4
• Level: $69$
• Weight: $4$
• Character: 69.16
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	69.16
Dual form			69.4.e.b.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.56552 - 1.80671i) q^{2} +(-2.52376 + 1.62192i) q^{3} +(0.325181 + 2.26169i) q^{4} +(1.13914 + 2.49437i) q^{5} +(-1.02066 + 7.09886i) q^{6} +(33.1193 - 9.72471i) q^{7} +(20.6842 + 13.2929i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 4 q^{2} + 18 q^{3} - 28 q^{4} - 6 q^{5} + 21 q^{6} - 4 q^{7} - 52 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.56552	−	1.80671i	0.553496	−	0.638768i	−0.408198	−	0.912893i	\(-0.633843\pi\)
							0.961694	+	0.274125i	\(0.0883883\pi\)
\(3\)	−2.52376	+	1.62192i	−0.485698	+	0.312139i
							\(5\)	1.13914	+	2.49437i	0.101888	+	0.223103i	0.953710	−	0.300728i	\(-0.0972296\pi\)
−0.851822	+	0.523831i	\(0.824502\pi\)
\(7\)	33.1193	−	9.72471i	1.78827	−	0.525085i	0.791938	−	0.610602i	\(-0.209072\pi\)
							0.996337	+	0.0855168i	\(0.0272541\pi\)
\(11\)	22.9049	+	26.4337i	0.627827	+	0.724550i	0.977173	−	0.212443i	\(-0.0681420\pi\)
							−0.349347	+	0.936993i	\(0.613597\pi\)
\(13\)	−66.4661	−	19.5162i	−1.41803	−	0.416371i	−0.519193	−	0.854657i	\(-0.673767\pi\)
							−0.898835	+	0.438286i	\(0.855586\pi\)
\(17\)	−11.0047	+	76.5392i	−0.157001	+	1.09197i	0.747119	+	0.664691i	\(0.231437\pi\)
							−0.904120	+	0.427278i	\(0.859472\pi\)
\(19\)	−4.78671	−	33.2923i	−0.0577972	−	0.401988i	−0.998098	−	0.0616442i	\(-0.980366\pi\)
							0.940301	−	0.340344i	\(-0.110543\pi\)
\(23\)	−58.3188	−	93.6265i	−0.528709	−	0.848803i
							\(29\)	−11.8660	+	82.5298i	−0.0759814	+	0.528462i	0.915911	+	0.401381i	\(0.131470\pi\)
−0.991892	+	0.127081i	\(0.959439\pi\)
\(31\)	−100.760	−	64.7544i	−0.583774	−	0.375169i	0.215168	−	0.976577i	\(-0.430970\pi\)
							−0.798942	+	0.601408i	\(0.794606\pi\)
\(37\)	122.732	−	268.746i	0.545326	−	1.19410i	−0.413605	−	0.910456i	\(-0.635731\pi\)
							0.958931	−	0.283640i	\(-0.0915421\pi\)
\(41\)	−111.329	−	243.777i	−0.424066	−	0.928575i	−0.994253	−	0.107060i	\(-0.965856\pi\)
							0.570186	−	0.821515i	\(-0.306871\pi\)
\(43\)	−279.664	+	179.729i	−0.991823	+	0.637406i	−0.932628	−	0.360840i	\(-0.882490\pi\)
							−0.0591957	+	0.998246i	\(0.518854\pi\)
\(47\)	−579.877			−1.79965			−0.899827	−	0.436246i	\(-0.856308\pi\)
							−0.899827	+	0.436246i	\(0.856308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.b.16.4	yes	60
3.2	odd	2		207.4.i.b.154.3		60
23.6	even	11		1587.4.a.w.1.10		30
23.13	even	11	inner	69.4.e.b.13.4	✓	60
23.17	odd	22		1587.4.a.v.1.10		30
69.59	odd	22		207.4.i.b.82.3		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.b.13.4	✓	60	23.13	even	11	inner
69.4.e.b.16.4	yes	60	1.1	even	1	trivial
207.4.i.b.82.3		60	69.59	odd	22
207.4.i.b.154.3		60	3.2	odd	2
1587.4.a.v.1.10		30	23.17	odd	22
1587.4.a.w.1.10		30	23.6	even	11