Embedded newform 129.2.n.b.62.1

• Label: 129.2.n.b.62.1
• Level: $129$
• Weight: $2$
• Character: 129.62
• 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			62.1
Character	\(\chi\)	\(=\)	129.62
Dual form			129.2.n.b.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71093 + 2.14544i) q^{2} +(1.69653 - 0.348970i) q^{3} +(-1.23059 - 5.39157i) q^{4} +(1.41668 - 0.965879i) q^{5} +(-2.15396 + 4.23688i) q^{6} +(2.06639 - 1.19303i) q^{7} +(8.72804 + 4.20320i) q^{8} +(2.75644 - 1.18408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71093	+	2.14544i	−1.20981	+	1.51706i	−0.415449	+	0.909616i	\(0.636376\pi\)
							−0.794364	+	0.607442i	\(0.792196\pi\)
\(3\)	1.69653	−	0.348970i	0.979493	−	0.201478i
							\(5\)	1.41668	−	0.965879i	0.633560	−	0.431954i	−0.203492	−	0.979077i	\(-0.565229\pi\)
0.837053	+	0.547122i	\(0.184277\pi\)
\(7\)	2.06639	−	1.19303i	0.781022	−	0.450923i	−0.0557706	−	0.998444i	\(-0.517762\pi\)
							0.836792	+	0.547521i	\(0.184428\pi\)
\(11\)	−4.70304	−	1.07344i	−1.41802	−	0.323654i	−0.556276	−	0.830998i	\(-0.687770\pi\)
							−0.861744	+	0.507344i	\(0.830627\pi\)
\(13\)	0.0641224	+	0.855654i	0.0177844	+	0.237316i	0.999011	+	0.0444587i	\(0.0141563\pi\)
							−0.981227	+	0.192857i	\(0.938225\pi\)
\(17\)	−0.871073	+	1.27763i	−0.211266	+	0.309871i	−0.917215	−	0.398392i	\(-0.869568\pi\)
							0.705949	+	0.708263i	\(0.250521\pi\)
\(19\)	−1.14291	+	3.70523i	−0.262202	+	0.850038i	0.724748	+	0.689014i	\(0.241956\pi\)
							−0.986950	+	0.161025i	\(0.948520\pi\)
\(23\)	−0.796623	+	0.858555i	−0.166107	+	0.179021i	−0.810620	−	0.585572i	\(-0.800870\pi\)
							0.644513	+	0.764594i	\(0.277060\pi\)
\(29\)	2.40321	+	0.362226i	0.446266	+	0.0672637i	0.368331	−	0.929695i	\(-0.379929\pi\)
							0.0779347	+	0.996958i	\(0.475167\pi\)
\(31\)	0.631718	+	1.60959i	0.113460	+	0.289091i	0.976313	−	0.216364i	\(-0.0694198\pi\)
							−0.862853	+	0.505455i	\(0.831325\pi\)
\(37\)	−4.41710	−	2.55021i	−0.726166	−	0.419252i	0.0908518	−	0.995864i	\(-0.471041\pi\)
							−0.817018	+	0.576612i	\(0.804374\pi\)
\(41\)	2.19036	+	1.74675i	0.342076	+	0.272797i	0.779425	−	0.626495i	\(-0.215511\pi\)
							−0.437349	+	0.899292i	\(0.644083\pi\)
\(43\)	6.40983	+	1.38350i	0.977490	+	0.210982i
							\(47\)	−8.16859	+	1.86443i	−1.19151	+	0.271955i	−0.771897	−	0.635748i	\(-0.780692\pi\)
−0.419614	+	0.907703i	\(0.637835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.n.b.62.1	✓	144
3.2	odd	2	inner	129.2.n.b.62.12	yes	144
43.34	odd	42	inner	129.2.n.b.77.12	yes	144
129.77	even	42	inner	129.2.n.b.77.1	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.n.b.62.1	✓	144	1.1	even	1	trivial
129.2.n.b.62.12	yes	144	3.2	odd	2	inner
129.2.n.b.77.1	yes	144	129.77	even	42	inner
129.2.n.b.77.12	yes	144	43.34	odd	42	inner