Embedded newform 69.2.g.a.56.1

• Label: 69.2.g.a.56.1
• 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			56.1
Character	\(\chi\)	\(=\)	69.56
Dual form			69.2.g.a.53.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99001 + 1.72435i) q^{2} +(0.319930 - 1.70225i) q^{3} +(0.702110 - 4.88329i) q^{4} +(1.01286 - 2.21786i) q^{5} +(2.29861 + 3.93915i) q^{6} +(-0.333512 + 1.13584i) q^{7} +(4.17610 + 6.49815i) q^{8} +(-2.79529 - 1.08920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99001	+	1.72435i	−1.40715	+	1.21930i	−0.464471	+	0.885589i	\(0.653755\pi\)
							−0.942676	+	0.333710i	\(0.891699\pi\)
\(3\)	0.319930	−	1.70225i	0.184712	−	0.982793i
							\(5\)	1.01286	−	2.21786i	0.452966	−	0.991857i	−0.536069	−	0.844174i	\(-0.680091\pi\)
0.989035	−	0.147683i	\(-0.0471814\pi\)
\(7\)	−0.333512	+	1.13584i	−0.126056	+	0.429306i	−0.998202	−	0.0599387i	\(-0.980909\pi\)
							0.872146	+	0.489245i	\(0.162728\pi\)
\(11\)	1.91217	−	2.20677i	0.576542	−	0.665365i	−0.390316	−	0.920681i	\(-0.627634\pi\)
							0.966858	+	0.255316i	\(0.0821795\pi\)
\(13\)	2.12617	−	0.624299i	0.589693	−	0.173149i	0.0267456	−	0.999642i	\(-0.491486\pi\)
							0.562947	+	0.826493i	\(0.309667\pi\)
\(17\)	0.460586	+	3.20345i	0.111709	+	0.776951i	0.966257	+	0.257579i	\(0.0829248\pi\)
							−0.854549	+	0.519371i	\(0.826166\pi\)
\(19\)	−2.06620	−	0.297075i	−0.474019	−	0.0681537i	−0.0988349	−	0.995104i	\(-0.531512\pi\)
							−0.375185	+	0.926950i	\(0.622421\pi\)
\(23\)	−2.40092	+	4.15158i	−0.500626	+	0.865664i
							\(29\)	3.98592	−	0.573089i	0.740167	−	0.106420i	0.238090	−	0.971243i	\(-0.423479\pi\)
0.502077	+	0.864823i	\(0.332570\pi\)
\(31\)	−3.31394	+	2.12974i	−0.595201	+	0.382512i	−0.803282	−	0.595599i	\(-0.796915\pi\)
							0.208081	+	0.978112i	\(0.433278\pi\)
\(37\)	9.07020	−	4.14222i	1.49113	−	0.680977i	0.507580	−	0.861605i	\(-0.330540\pi\)
							0.983551	+	0.180628i	\(0.0578131\pi\)
\(41\)	−1.04401	−	0.476785i	−0.163048	−	0.0744614i	0.332219	−	0.943202i	\(-0.392203\pi\)
							−0.495267	+	0.868741i	\(0.664930\pi\)
\(43\)	−5.65476	+	8.79897i	−0.862342	+	1.34183i	0.0763522	+	0.997081i	\(0.475673\pi\)
							−0.938695	+	0.344750i	\(0.887964\pi\)
\(47\)			3.92905i			0.573111i	0.958064	+	0.286556i	\(0.0925103\pi\)
							−0.958064	+	0.286556i	\(0.907490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.56.1	yes	60
3.2	odd	2	inner	69.2.g.a.56.6	yes	60
23.7	odd	22	inner	69.2.g.a.53.6	yes	60
69.53	even	22	inner	69.2.g.a.53.1	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.53.1	✓	60	69.53	even	22	inner
69.2.g.a.53.6	yes	60	23.7	odd	22	inner
69.2.g.a.56.1	yes	60	1.1	even	1	trivial
69.2.g.a.56.6	yes	60	3.2	odd	2	inner