Embedded newform 129.2.j.a.2.9

• Label: 129.2.j.a.2.9
• Level: $129$
• Weight: $2$
• Character: 129.2
• 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			2.9
Character	\(\chi\)	\(=\)	129.2
Dual form			129.2.j.a.65.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.754176 - 0.363192i) q^{2} +(-1.36453 + 1.06679i) q^{3} +(-0.810106 + 1.01584i) q^{4} +(-0.252915 + 1.10809i) q^{5} +(-0.641646 + 1.30014i) q^{6} +2.80770i q^{7} +(-0.614550 + 2.69252i) q^{8} +(0.723897 - 2.91135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.754176	−	0.363192i	0.533283	−	0.256816i	−0.147806	−	0.989016i	\(-0.547221\pi\)
							0.681089	+	0.732201i	\(0.261507\pi\)
\(3\)	−1.36453	+	1.06679i	−0.787813	+	0.615914i
							\(5\)	−0.252915	+	1.10809i	−0.113107	+	0.495555i	0.886362	+	0.462992i	\(0.153224\pi\)
−0.999470	+	0.0325631i	\(0.989633\pi\)
\(7\)			2.80770i			1.06121i	0.847619	+	0.530605i	\(0.178035\pi\)
							−0.847619	+	0.530605i	\(0.821965\pi\)
\(11\)	4.14007	−	3.30159i	1.24828	−	0.995467i	0.248638	−	0.968597i	\(-0.420017\pi\)
							0.999639	−	0.0268708i	\(-0.00855427\pi\)
\(13\)	0.0697816	−	0.305733i	0.0193539	−	0.0847951i	−0.964329	−	0.264708i	\(-0.914724\pi\)
							0.983683	+	0.179913i	\(0.0575816\pi\)
\(17\)	−6.26659	+	1.43031i	−1.51987	+	0.346901i	−0.899329	−	0.437272i	\(-0.855945\pi\)
							−0.620543	+	0.784173i	\(0.713088\pi\)
\(19\)	3.39643	+	2.70856i	0.779194	+	0.621386i	0.930161	−	0.367152i	\(-0.119667\pi\)
							−0.150967	+	0.988539i	\(0.548239\pi\)
\(23\)	4.35503	−	3.47302i	0.908087	−	0.724175i	−0.0535311	−	0.998566i	\(-0.517048\pi\)
							0.961618	+	0.274391i	\(0.0884762\pi\)
\(29\)	−3.75785	+	1.80968i	−0.697815	+	0.336050i	−0.748936	−	0.662643i	\(-0.769435\pi\)
							0.0511210	+	0.998692i	\(0.483721\pi\)
\(31\)	8.10571	−	3.90351i	1.45583	−	0.701090i	0.472233	−	0.881474i	\(-0.343448\pi\)
							0.983596	+	0.180383i	\(0.0577339\pi\)
\(37\)		−	1.67365i		−	0.275146i	−0.990492	−	0.137573i	\(-0.956070\pi\)
							0.990492	−	0.137573i	\(-0.0439301\pi\)
\(41\)	−1.16608	−	2.42140i	−0.182112	−	0.378159i	0.789849	−	0.613301i	\(-0.210159\pi\)
							−0.971961	+	0.235142i	\(0.924444\pi\)
\(43\)	−6.12226	+	2.34902i	−0.933636	+	0.358223i
							\(47\)	2.76053	+	2.20145i	0.402665	+	0.321115i	0.803795	−	0.594906i	\(-0.202811\pi\)
−0.401130	+	0.916021i	\(0.631382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.j.a.2.9	yes	72
3.2	odd	2	inner	129.2.j.a.2.4	✓	72
43.22	odd	14	inner	129.2.j.a.65.4	yes	72
129.65	even	14	inner	129.2.j.a.65.9	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.j.a.2.4	✓	72	3.2	odd	2	inner
129.2.j.a.2.9	yes	72	1.1	even	1	trivial
129.2.j.a.65.4	yes	72	43.22	odd	14	inner
129.2.j.a.65.9	yes	72	129.65	even	14	inner