Embedded newform 69.4.g.a.5.9

• Label: 69.4.g.a.5.9
• Level: $69$
• Weight: $4$
• Character: 69.5
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	69.5
Dual form			69.4.g.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222958 + 0.759325i) q^{2} +(4.12739 + 3.15668i) q^{3} +(6.20316 + 3.98653i) q^{4} +(2.75670 - 19.1733i) q^{5} +(-3.31718 + 2.43022i) q^{6} +(15.1346 - 6.91176i) q^{7} +(-9.19481 + 7.96735i) q^{8} +(7.07070 + 26.0577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 5 q^{3} + 50 q^{4} - 32 q^{6} - 22 q^{7} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222958	+	0.759325i	−0.0788275	+	0.268462i	−0.989471	−	0.144731i	\(-0.953768\pi\)
							0.910644	+	0.413193i	\(0.135586\pi\)
\(3\)	4.12739	+	3.15668i	0.794317	+	0.607504i
							\(5\)	2.75670	−	19.1733i	0.246567	−	1.71491i	−0.371203	−	0.928552i	\(-0.621055\pi\)
0.617770	−	0.786359i	\(-0.288036\pi\)
\(7\)	15.1346	−	6.91176i	0.817194	−	0.373200i	0.0374500	−	0.999299i	\(-0.488077\pi\)
							0.779744	+	0.626099i	\(0.215349\pi\)
\(11\)	−41.2300	+	12.1062i	−1.13012	+	0.331833i	−0.792754	−	0.609542i	\(-0.791353\pi\)
							−0.337364	+	0.941374i	\(0.609535\pi\)
\(13\)	0.290001	−	0.635014i	0.00618707	−	0.0135478i	−0.906513	−	0.422177i	\(-0.861266\pi\)
							0.912700	+	0.408630i	\(0.133993\pi\)
\(17\)	26.8186	−	17.2353i	0.382616	−	0.245893i	−0.335168	−	0.942159i	\(-0.608793\pi\)
							0.717784	+	0.696266i	\(0.245157\pi\)
\(19\)	−44.3460	+	69.0038i	−0.535457	+	0.833187i	−0.998589	−	0.0531021i	\(-0.983089\pi\)
							0.463132	+	0.886289i	\(0.346725\pi\)
\(23\)	−98.8113	−	49.0237i	−0.895808	−	0.444441i
							\(29\)	39.2593	+	61.0886i	0.251388	+	0.391168i	0.943896	−	0.330243i	\(-0.107131\pi\)
−0.692508	+	0.721410i	\(0.743494\pi\)
\(31\)	−167.243	−	193.008i	−0.968958	−	1.11824i	−0.992951	−	0.118525i	\(-0.962184\pi\)
							0.0239935	−	0.999712i	\(-0.492362\pi\)
\(37\)	366.309	−	52.6673i	1.62759	−	0.234012i	0.732783	−	0.680463i	\(-0.238221\pi\)
							0.894808	+	0.446451i	\(0.147312\pi\)
\(41\)	224.751	+	32.3143i	0.856102	+	0.123089i	0.556370	−	0.830935i	\(-0.312194\pi\)
							0.299732	+	0.954024i	\(0.403103\pi\)
\(43\)	−343.682	−	297.802i	−1.21886	−	1.05615i	−0.996703	−	0.0811384i	\(-0.974144\pi\)
							−0.222158	−	0.975011i	\(-0.571310\pi\)
\(47\)			95.5076i			0.296409i	0.988957	+	0.148204i	\(0.0473494\pi\)
							−0.988957	+	0.148204i	\(0.952651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.g.a.5.9	✓	220
3.2	odd	2	inner	69.4.g.a.5.14	yes	220
23.14	odd	22	inner	69.4.g.a.14.14	yes	220
69.14	even	22	inner	69.4.g.a.14.9	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.g.a.5.9	✓	220	1.1	even	1	trivial
69.4.g.a.5.14	yes	220	3.2	odd	2	inner
69.4.g.a.14.9	yes	220	69.14	even	22	inner
69.4.g.a.14.14	yes	220	23.14	odd	22	inner