Embedded newform 69.4.g.a.5.15

• Label: 69.4.g.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	69.5
Dual form			69.4.g.a.14.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.579088 - 1.97219i) q^{2} +(-4.53927 + 2.52884i) q^{3} +(3.17583 + 2.04098i) q^{4} +(1.18604 - 8.24906i) q^{5} +(2.35872 + 10.4167i) q^{6} +(14.0781 - 6.42924i) q^{7} +(18.2916 - 15.8497i) q^{8} +(14.2100 - 22.9582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 5 q^{3} + 50 q^{4} - 32 q^{6} - 22 q^{7} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.579088	−	1.97219i	0.204739	−	0.697275i	−0.791543	−	0.611114i	\(-0.790722\pi\)
							0.996281	−	0.0861615i	\(-0.0274601\pi\)
\(3\)	−4.53927	+	2.52884i	−0.873583	+	0.486675i
							\(5\)	1.18604	−	8.24906i	0.106082	−	0.737818i	−0.865465	−	0.500970i	\(-0.832977\pi\)
0.971547	−	0.236848i	\(-0.0761143\pi\)
\(7\)	14.0781	−	6.42924i	0.760144	−	0.347146i	0.00265728	−	0.999996i	\(-0.499154\pi\)
							0.757487	+	0.652850i	\(0.226427\pi\)
\(11\)	19.3548	−	5.68309i	0.530518	−	0.155774i	−0.00549319	−	0.999985i	\(-0.501749\pi\)
							0.536011	+	0.844211i	\(0.319930\pi\)
\(13\)	5.90018	−	12.9196i	0.125878	−	0.275635i	−0.836192	−	0.548437i	\(-0.815223\pi\)
							0.962070	+	0.272802i	\(0.0879504\pi\)
\(17\)	5.18575	−	3.33268i	0.0739840	−	0.0475467i	−0.503126	−	0.864213i	\(-0.667817\pi\)
							0.577110	+	0.816666i	\(0.304180\pi\)
\(19\)	−21.4298	+	33.3454i	−0.258754	+	0.402629i	−0.946188	−	0.323618i	\(-0.895101\pi\)
							0.687434	+	0.726247i	\(0.258737\pi\)
\(23\)	−110.217	+	4.39292i	−0.999207	+	0.0398256i
							\(29\)	−27.2150	−	42.3474i	−0.174266	−	0.271163i	0.743123	−	0.669155i	\(-0.233344\pi\)
−0.917389	+	0.397992i	\(0.869707\pi\)
\(31\)	162.426	+	187.450i	0.941052	+	1.08603i	0.996160	+	0.0875513i	\(0.0279042\pi\)
							−0.0551082	+	0.998480i	\(0.517550\pi\)
\(37\)	−111.718	+	16.0626i	−0.496386	+	0.0713695i	−0.385961	−	0.922515i	\(-0.626130\pi\)
							−0.110424	+	0.993885i	\(0.535221\pi\)
\(41\)	−414.521	−	59.5991i	−1.57896	−	0.227020i	−0.703636	−	0.710560i	\(-0.748442\pi\)
							−0.875322	+	0.483540i	\(0.839351\pi\)
\(43\)	−152.298	−	131.967i	−0.540121	−	0.468017i	0.341562	−	0.939859i	\(-0.389044\pi\)
							−0.881683	+	0.471842i	\(0.843589\pi\)
\(47\)			180.055i			0.558801i	0.960175	+	0.279401i	\(0.0901358\pi\)
							−0.960175	+	0.279401i	\(0.909864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.g.a.5.15	yes	220
3.2	odd	2	inner	69.4.g.a.5.8	✓	220
23.14	odd	22	inner	69.4.g.a.14.8	yes	220
69.14	even	22	inner	69.4.g.a.14.15	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.g.a.5.8	✓	220	3.2	odd	2	inner
69.4.g.a.5.15	yes	220	1.1	even	1	trivial
69.4.g.a.14.8	yes	220	23.14	odd	22	inner
69.4.g.a.14.15	yes	220	69.14	even	22	inner