Embedded newform 129.2.n.b.77.8

• Label: 129.2.n.b.77.8
• Level: $129$
• Weight: $2$
• Character: 129.77
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.n (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			77.8
Character	\(\chi\)	\(=\)	129.77
Dual form			129.2.n.b.62.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.617756 + 0.774642i) q^{2} +(-0.339510 - 1.69845i) q^{3} +(0.226595 - 0.992775i) q^{4} +(-1.92344 - 1.31138i) q^{5} +(1.10596 - 1.31223i) q^{6} +(1.23634 + 0.713802i) q^{7} +(2.69439 - 1.29755i) q^{8} +(-2.76947 + 1.15328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.617756	+	0.774642i	0.436820	+	0.547755i	0.950702	−	0.310106i	\(-0.100365\pi\)
							−0.513882	+	0.857861i	\(0.671793\pi\)
\(3\)	−0.339510	−	1.69845i	−0.196016	−	0.980601i
							\(5\)	−1.92344	−	1.31138i	−0.860186	−	0.586465i	0.0508892	−	0.998704i	\(-0.483794\pi\)
−0.911076	+	0.412239i	\(0.864747\pi\)
\(7\)	1.23634	+	0.713802i	0.467293	+	0.269792i	0.715106	−	0.699016i	\(-0.246379\pi\)
							−0.247813	+	0.968808i	\(0.579712\pi\)
\(11\)	2.11544	−	0.482836i	0.637830	−	0.145581i	0.108636	−	0.994082i	\(-0.465352\pi\)
							0.529194	+	0.848501i	\(0.322494\pi\)
\(13\)	−0.101781	+	1.35818i	−0.0282291	+	0.376691i	0.965141	+	0.261730i	\(0.0842931\pi\)
							−0.993370	+	0.114960i	\(0.963326\pi\)
\(17\)	2.52256	+	3.69992i	0.611811	+	0.897362i	0.999704	−	0.0243424i	\(-0.00774921\pi\)
							−0.387893	+	0.921705i	\(0.626797\pi\)
\(19\)	1.81102	+	5.87119i	0.415477	+	1.34694i	0.885643	+	0.464366i	\(0.153718\pi\)
							−0.470166	+	0.882578i	\(0.655806\pi\)
\(23\)	−3.49006	−	3.76139i	−0.727727	−	0.784303i	0.255477	−	0.966815i	\(-0.417768\pi\)
							−0.983204	+	0.182512i	\(0.941577\pi\)
\(29\)	0.419807	−	0.0632757i	0.0779562	−	0.0117500i	−0.109948	−	0.993937i	\(-0.535069\pi\)
							0.187905	+	0.982187i	\(0.439830\pi\)
\(31\)	−0.183919	+	0.468618i	−0.0330328	+	0.0841663i	−0.946438	−	0.322884i	\(-0.895348\pi\)
							0.913406	+	0.407051i	\(0.133443\pi\)
\(37\)	2.80646	−	1.62031i	0.461379	−	0.266377i	−0.251245	−	0.967924i	\(-0.580840\pi\)
							0.712624	+	0.701546i	\(0.247507\pi\)
\(41\)	−2.01013	+	1.60302i	−0.313929	+	0.250350i	−0.767760	−	0.640738i	\(-0.778629\pi\)
							0.453831	+	0.891088i	\(0.350057\pi\)
\(43\)	−0.712060	+	6.51866i	−0.108588	+	0.994087i
							\(47\)	−9.94292	−	2.26941i	−1.45032	−	0.331027i	−0.576431	−	0.817146i	\(-0.695555\pi\)
−0.873893	+	0.486118i	\(0.838412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.n.b.77.8	yes	144
3.2	odd	2	inner	129.2.n.b.77.5	yes	144
43.19	odd	42	inner	129.2.n.b.62.5	✓	144
129.62	even	42	inner	129.2.n.b.62.8	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.n.b.62.5	✓	144	43.19	odd	42	inner
129.2.n.b.62.8	yes	144	129.62	even	42	inner
129.2.n.b.77.5	yes	144	3.2	odd	2	inner
129.2.n.b.77.8	yes	144	1.1	even	1	trivial