Embedded newform 69.4.e.a.16.5

• Label: 69.4.e.a.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	69.16
Dual form			69.4.e.a.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38600 - 1.59953i) q^{2} +(2.52376 - 1.62192i) q^{3} +(0.501022 + 3.48468i) q^{4} +(6.21106 + 13.6003i) q^{5} +(0.903619 - 6.28481i) q^{6} +(10.6195 - 3.11816i) q^{7} +(20.5122 + 13.1824i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 18 q^{3} - 28 q^{4} + 22 q^{5} - 33 q^{6} + 24 q^{7} + 16 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38600	−	1.59953i	0.490025	−	0.565519i	−0.455848	−	0.890058i	\(-0.650664\pi\)
							0.945872	+	0.324539i	\(0.105209\pi\)
\(3\)	2.52376	−	1.62192i	0.485698	−	0.312139i
							\(5\)	6.21106	+	13.6003i	0.555534	+	1.21645i	0.954149	+	0.299332i	\(0.0967639\pi\)
−0.398614	+	0.917119i	\(0.630509\pi\)
\(7\)	10.6195	−	3.11816i	0.573398	−	0.168365i	0.0178373	−	0.999841i	\(-0.494322\pi\)
							0.555561	+	0.831476i	\(0.312504\pi\)
\(11\)	−39.9561	−	46.1118i	−1.09520	−	1.26393i	−0.962062	−	0.272831i	\(-0.912040\pi\)
							−0.133139	−	0.991097i	\(-0.542506\pi\)
\(13\)	−36.3835	−	10.6832i	−0.776229	−	0.227921i	−0.130459	−	0.991454i	\(-0.541645\pi\)
							−0.645770	+	0.763532i	\(0.723463\pi\)
\(17\)	13.5607	−	94.3170i	0.193468	−	1.34560i	−0.629272	−	0.777185i	\(-0.716647\pi\)
							0.822740	−	0.568417i	\(-0.192444\pi\)
\(19\)	15.7424	+	109.491i	0.190082	+	1.32205i	0.831787	+	0.555095i	\(0.187318\pi\)
							−0.641705	+	0.766951i	\(0.721773\pi\)
\(23\)	44.8621	+	100.769i	0.406713	+	0.913556i
							\(29\)	29.6081	−	205.929i	0.189589	−	1.31862i	−0.643483	−	0.765460i	\(-0.722511\pi\)
0.833073	−	0.553163i	\(-0.186579\pi\)
\(31\)	−140.048	−	90.0031i	−0.811396	−	0.521453i	0.0679200	−	0.997691i	\(-0.478364\pi\)
							−0.879316	+	0.476238i	\(0.842000\pi\)
\(37\)	−50.3233	+	110.193i	−0.223597	+	0.489610i	−0.987870	−	0.155283i	\(-0.950371\pi\)
							0.764273	+	0.644893i	\(0.223098\pi\)
\(41\)	−173.108	−	379.054i	−0.659388	−	1.44386i	−0.883091	−	0.469202i	\(-0.844542\pi\)
							0.223702	−	0.974657i	\(-0.428186\pi\)
\(43\)	−201.188	+	129.296i	−0.713510	+	0.458545i	−0.846374	−	0.532589i	\(-0.821219\pi\)
							0.132864	+	0.991134i	\(0.457583\pi\)
\(47\)	41.4943			0.128778			0.0643890	−	0.997925i	\(-0.479490\pi\)
							0.0643890	+	0.997925i	\(0.479490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.a.16.5	yes	60
3.2	odd	2		207.4.i.c.154.2		60
23.6	even	11		1587.4.a.t.1.10		30
23.13	even	11	inner	69.4.e.a.13.5	✓	60
23.17	odd	22		1587.4.a.u.1.10		30
69.59	odd	22		207.4.i.c.82.2		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.a.13.5	✓	60	23.13	even	11	inner
69.4.e.a.16.5	yes	60	1.1	even	1	trivial
207.4.i.c.82.2		60	69.59	odd	22
207.4.i.c.154.2		60	3.2	odd	2
1587.4.a.t.1.10		30	23.6	even	11
1587.4.a.u.1.10		30	23.17	odd	22