Embedded newform 69.4.e.b.16.5

• Label: 69.4.e.b.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.b.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.56668 - 1.80804i) q^{2} +(-2.52376 + 1.62192i) q^{3} +(0.323978 + 2.25332i) q^{4} +(6.20398 + 13.5848i) q^{5} +(-1.02142 + 7.10410i) q^{6} +(-24.5496 + 7.20842i) q^{7} +(20.6825 + 13.2918i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 4 q^{2} + 18 q^{3} - 28 q^{4} - 6 q^{5} + 21 q^{6} - 4 q^{7} - 52 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.56668	−	1.80804i	0.553905	−	0.639240i	−0.407884	−	0.913034i	\(-0.633733\pi\)
							0.961788	+	0.273794i	\(0.0882786\pi\)
\(3\)	−2.52376	+	1.62192i	−0.485698	+	0.312139i
							\(5\)	6.20398	+	13.5848i	0.554901	+	1.21506i	0.954455	+	0.298354i	\(0.0964376\pi\)
−0.399554	+	0.916709i	\(0.630835\pi\)
\(7\)	−24.5496	+	7.20842i	−1.32555	+	0.389218i	−0.866494	−	0.499187i	\(-0.833632\pi\)
							−0.459060	+	0.888405i	\(0.651814\pi\)
\(11\)	−11.9090	−	13.7437i	−0.326427	−	0.376716i	0.568687	−	0.822554i	\(-0.307451\pi\)
							−0.895114	+	0.445837i	\(0.852906\pi\)
\(13\)	86.3921	+	25.3670i	1.84314	+	0.541196i	0.999994	+	0.00351590i	\(0.00111915\pi\)
							0.843149	+	0.537680i	\(0.180699\pi\)
\(17\)	0.0854087	−	0.594031i	0.00121851	−	0.00847492i	−0.989203	−	0.146551i	\(-0.953183\pi\)
							0.990422	+	0.138076i	\(0.0440918\pi\)
\(19\)	−13.3171	−	92.6227i	−0.160798	−	1.11837i	−0.897134	−	0.441758i	\(-0.854355\pi\)
							0.736336	−	0.676616i	\(-0.236554\pi\)
\(23\)	−110.140	−	6.01038i	−0.998514	−	0.0544891i
							\(29\)	−10.2601	+	71.3608i	−0.0656986	+	0.456944i	0.930243	+	0.366944i	\(0.119596\pi\)
−0.995942	+	0.0900001i	\(0.971313\pi\)
\(31\)	219.303	+	140.937i	1.27058	+	0.816551i	0.989695	−	0.143191i	\(-0.0457363\pi\)
							0.280883	+	0.959742i	\(0.409373\pi\)
\(37\)	53.6514	−	117.480i	0.238385	−	0.521990i	−0.752193	−	0.658943i	\(-0.771004\pi\)
							0.990577	+	0.136953i	\(0.0437311\pi\)
\(41\)	−93.5789	−	204.909i	−0.356453	−	0.780523i	−0.999887	−	0.0150215i	\(-0.995218\pi\)
							0.643434	−	0.765501i	\(-0.277509\pi\)
\(43\)	396.280	−	254.674i	1.40540	−	0.903195i	0.405458	−	0.914113i	\(-0.367112\pi\)
							0.999940	+	0.0109185i	\(0.00347553\pi\)
\(47\)	194.164			0.602589			0.301294	−	0.953531i	\(-0.402581\pi\)
							0.301294	+	0.953531i	\(0.402581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.b.16.5	yes	60
3.2	odd	2		207.4.i.b.154.2		60
23.6	even	11		1587.4.a.w.1.9		30
23.13	even	11	inner	69.4.e.b.13.5	✓	60
23.17	odd	22		1587.4.a.v.1.9		30
69.59	odd	22		207.4.i.b.82.2		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.b.13.5	✓	60	23.13	even	11	inner
69.4.e.b.16.5	yes	60	1.1	even	1	trivial
207.4.i.b.82.2		60	69.59	odd	22
207.4.i.b.154.2		60	3.2	odd	2
1587.4.a.v.1.9		30	23.17	odd	22
1587.4.a.w.1.9		30	23.6	even	11