Embedded newform 129.2.j.a.32.1

• Label: 129.2.j.a.32.1
• Level: $129$
• Weight: $2$
• Character: 129.32
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			32.1
Character	\(\chi\)	\(=\)	129.32
Dual form			129.2.j.a.125.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54397 - 1.93608i) q^{2} +(-0.980767 + 1.42762i) q^{3} +(-0.919509 + 4.02863i) q^{4} +(2.61651 - 1.26004i) q^{5} +(4.27825 - 0.305359i) q^{6} -2.05995i q^{7} +(4.75723 - 2.29096i) q^{8} +(-1.07619 - 2.80032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54397	−	1.93608i	−1.09175	−	1.36901i	−0.923647	−	0.383244i	\(-0.874807\pi\)
							−0.168104	−	0.985769i	\(-0.553765\pi\)
\(3\)	−0.980767	+	1.42762i	−0.566246	+	0.824236i
							\(5\)	2.61651	−	1.26004i	1.17014	−	0.563508i	0.255115	−	0.966911i	\(-0.417887\pi\)
0.915023	+	0.403402i	\(0.132172\pi\)
\(7\)		−	2.05995i		−	0.778588i	−0.921114	−	0.389294i	\(-0.872719\pi\)
							0.921114	−	0.389294i	\(-0.127281\pi\)
\(11\)	−0.179505	+	0.0409708i	−0.0541227	+	0.0123532i	−0.249497	−	0.968376i	\(-0.580265\pi\)
							0.195374	+	0.980729i	\(0.437408\pi\)
\(13\)	5.16952	−	2.48951i	1.43377	−	0.690466i	0.454073	−	0.890965i	\(-0.349971\pi\)
							0.979694	+	0.200499i	\(0.0642563\pi\)
\(17\)	0.844195	−	1.75299i	0.204747	−	0.425162i	−0.773157	−	0.634215i	\(-0.781324\pi\)
							0.977904	+	0.209052i	\(0.0670379\pi\)
\(19\)	−6.46394	−	1.47535i	−1.48293	−	0.338469i	−0.596982	−	0.802255i	\(-0.703634\pi\)
							−0.885947	+	0.463786i	\(0.846491\pi\)
\(23\)	6.67986	−	1.52463i	1.39285	−	0.317908i	0.540702	−	0.841214i	\(-0.318159\pi\)
							0.852145	+	0.523306i	\(0.175302\pi\)
\(29\)	−0.595665	−	0.746941i	−0.110612	−	0.138703i	0.723443	−	0.690384i	\(-0.242558\pi\)
							−0.834056	+	0.551680i	\(0.813987\pi\)
\(31\)	3.43002	+	4.30111i	0.616050	+	0.772502i	0.987783	−	0.155838i	\(-0.0498077\pi\)
							−0.371733	+	0.928340i	\(0.621236\pi\)
\(37\)			7.76737i			1.27695i	0.769644	+	0.638473i	\(0.220434\pi\)
							−0.769644	+	0.638473i	\(0.779566\pi\)
\(41\)	−5.19173	+	4.14027i	−0.810812	+	0.646601i	−0.938526	−	0.345210i	\(-0.887808\pi\)
							0.127713	+	0.991811i	\(0.459236\pi\)
\(43\)	−6.06602	+	2.49066i	−0.925059	+	0.379822i
							\(47\)	−4.76151	−	1.08678i	−0.694537	−	0.158524i	−0.139343	−	0.990244i	\(-0.544499\pi\)
−0.555194	+	0.831721i	\(0.687356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.j.a.32.1	✓	72
3.2	odd	2	inner	129.2.j.a.32.12	yes	72
43.39	odd	14	inner	129.2.j.a.125.12	yes	72
129.125	even	14	inner	129.2.j.a.125.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.j.a.32.1	✓	72	1.1	even	1	trivial
129.2.j.a.32.12	yes	72	3.2	odd	2	inner
129.2.j.a.125.1	yes	72	129.125	even	14	inner
129.2.j.a.125.12	yes	72	43.39	odd	14	inner