Embedded newform 69.4.g.a.5.8

• Label: 69.4.g.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	69.5
Dual form			69.4.g.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.579088 + 1.97219i) q^{2} +(-1.85709 + 4.85296i) q^{3} +(3.17583 + 2.04098i) q^{4} +(-1.18604 + 8.24906i) q^{5} +(-8.49555 - 6.47283i) q^{6} +(14.0781 - 6.42924i) q^{7} +(-18.2916 + 15.8497i) q^{8} +(-20.1024 - 18.0248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 5 q^{3} + 50 q^{4} - 32 q^{6} - 22 q^{7} - 5 q^{9}+O(q^{10}) \)

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	\(0.209278\pi\)
							−0.996281	+	0.0861615i	\(0.972540\pi\)
\(3\)	−1.85709	+	4.85296i	−0.357398	+	0.933952i
							\(5\)	−1.18604	+	8.24906i	−0.106082	+	0.737818i	0.865465	+	0.500970i	\(0.167023\pi\)
−0.971547	+	0.236848i	\(0.923886\pi\)
\(7\)	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	−0.536011	−	0.844211i	\(-0.680070\pi\)
							0.00549319	+	0.999985i	\(0.498251\pi\)
\(13\)	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	−0.577110	−	0.816666i	\(-0.695820\pi\)
							0.503126	+	0.864213i	\(0.332183\pi\)
\(19\)	−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	0.917389	−	0.397992i	\(-0.130293\pi\)
−0.743123	+	0.669155i	\(0.766656\pi\)
\(31\)	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	0.875322	−	0.483540i	\(-0.160649\pi\)
							0.703636	+	0.710560i	\(0.251558\pi\)
\(43\)	−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	\(-0.909864\pi\)
							0.960175	−	0.279401i	\(-0.0901358\pi\)

(See \(a_n\) instead)

Twists

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

    

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