Embedded newform 69.2.g.a.56.4

• Label: 69.2.g.a.56.4
• Level: $69$
• Weight: $2$
• Character: 69.56
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			56.4
Character	\(\chi\)	\(=\)	69.56
Dual form			69.2.g.a.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.256861 - 0.222571i) q^{2} +(-0.399473 + 1.68535i) q^{3} +(-0.268190 + 1.86530i) q^{4} +(0.197534 - 0.432538i) q^{5} +(0.272503 + 0.521813i) q^{6} +(0.641364 - 2.18428i) q^{7} +(0.713777 + 1.11066i) q^{8} +(-2.68084 - 1.34651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{3}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.256861	−	0.222571i	0.181628	−	0.157382i	−0.559302	−	0.828964i	\(-0.688931\pi\)
							0.740930	+	0.671583i	\(0.234385\pi\)
\(3\)	−0.399473	+	1.68535i	−0.230636	+	0.973040i
							\(5\)	0.197534	−	0.432538i	0.0883397	−	0.193437i	−0.860306	−	0.509779i	\(-0.829727\pi\)
0.948645	+	0.316342i	\(0.102455\pi\)
\(7\)	0.641364	−	2.18428i	0.242413	−	0.825582i	−0.744952	−	0.667118i	\(-0.767528\pi\)
							0.987365	−	0.158464i	\(-0.0506542\pi\)
\(11\)	3.61237	−	4.16889i	1.08917	−	1.25697i	0.124865	−	0.992174i	\(-0.460150\pi\)
							0.964305	−	0.264795i	\(-0.0853043\pi\)
\(13\)	−0.876516	+	0.257368i	−0.243102	+	0.0713811i	−0.401014	−	0.916072i	\(-0.631342\pi\)
							0.157912	+	0.987453i	\(0.449524\pi\)
\(17\)	0.637350	+	4.43287i	0.154580	+	1.07513i	0.908416	+	0.418066i	\(0.137292\pi\)
							−0.753836	+	0.657062i	\(0.771799\pi\)
\(19\)	−3.96860	−	0.570598i	−0.910459	−	0.130904i	−0.328859	−	0.944379i	\(-0.606664\pi\)
							−0.581600	+	0.813475i	\(0.697573\pi\)
\(23\)	−2.58111	−	4.04201i	−0.538199	−	0.842818i
							\(29\)	−2.58818	+	0.372124i	−0.480613	+	0.0691017i	−0.378363	−	0.925657i	\(-0.623513\pi\)
−0.102249	+	0.994759i	\(0.532604\pi\)
\(31\)	2.56837	−	1.65059i	0.461292	−	0.296454i	−0.289278	−	0.957245i	\(-0.593415\pi\)
							0.750570	+	0.660791i	\(0.229779\pi\)
\(37\)	1.02265	−	0.467030i	0.168123	−	0.0767793i	−0.329576	−	0.944129i	\(-0.606906\pi\)
							0.497699	+	0.867350i	\(0.334178\pi\)
\(41\)	−5.22785	−	2.38748i	−0.816453	−	0.372862i	−0.0369947	−	0.999315i	\(-0.511778\pi\)
							−0.779459	+	0.626454i	\(0.784506\pi\)
\(43\)	−1.87281	+	2.91415i	−0.285601	+	0.444403i	−0.954178	−	0.299239i	\(-0.903267\pi\)
							0.668577	+	0.743642i	\(0.266904\pi\)
\(47\)			10.1848i			1.48560i	0.669513	+	0.742800i	\(0.266503\pi\)
							−0.669513	+	0.742800i	\(0.733497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.56.4	yes	60
3.2	odd	2	inner	69.2.g.a.56.3	yes	60
23.7	odd	22	inner	69.2.g.a.53.3	✓	60
69.53	even	22	inner	69.2.g.a.53.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.53.3	✓	60	23.7	odd	22	inner
69.2.g.a.53.4	yes	60	69.53	even	22	inner
69.2.g.a.56.3	yes	60	3.2	odd	2	inner
69.2.g.a.56.4	yes	60	1.1	even	1	trivial