Embedded newform 129.2.j.a.2.2

• Label: 129.2.j.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	129.2
Dual form			129.2.j.a.65.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79909 + 0.866396i) q^{2} +(1.60410 + 0.653342i) q^{3} +(1.23910 - 1.55378i) q^{4} +(-0.490759 + 2.15015i) q^{5} +(-3.45198 + 0.214367i) q^{6} +0.451945i q^{7} +(0.00561411 - 0.0245970i) q^{8} +(2.14629 + 2.09606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79909	+	0.866396i	−1.27215	+	0.612634i	−0.943361	−	0.331769i	\(-0.892354\pi\)
							−0.328788	+	0.944404i	\(0.606640\pi\)
\(3\)	1.60410	+	0.653342i	0.926129	+	0.377207i
							\(5\)	−0.490759	+	2.15015i	−0.219474	+	0.961578i	0.738394	+	0.674369i	\(0.235584\pi\)
−0.957868	+	0.287209i	\(0.907273\pi\)
\(7\)			0.451945i			0.170819i	0.996346	+	0.0854096i	\(0.0272199\pi\)
							−0.996346	+	0.0854096i	\(0.972780\pi\)
\(11\)	−0.584862	+	0.466412i	−0.176343	+	0.140629i	−0.707684	−	0.706530i	\(-0.750260\pi\)
							0.531341	+	0.847158i	\(0.321688\pi\)
\(13\)	−0.438260	+	1.92014i	−0.121552	+	0.532552i	0.877084	+	0.480337i	\(0.159485\pi\)
							−0.998636	+	0.0522156i	\(0.983372\pi\)
\(17\)	−5.03167	+	1.14845i	−1.22036	+	0.278539i	−0.783729	−	0.621102i	\(-0.786685\pi\)
							−0.436630	+	0.899641i	\(0.643828\pi\)
\(19\)	−1.09761	−	0.875313i	−0.251809	−	0.200811i	0.489448	−	0.872033i	\(-0.337198\pi\)
							−0.741256	+	0.671222i	\(0.765770\pi\)
\(23\)	6.31654	−	5.03728i	1.31709	−	1.05034i	0.322489	−	0.946573i	\(-0.395480\pi\)
							0.994601	−	0.103771i	\(-0.0330910\pi\)
\(29\)	8.19488	−	3.94644i	1.52175	−	0.732836i	0.528512	−	0.848926i	\(-0.322750\pi\)
							0.993239	+	0.116090i	\(0.0370360\pi\)
\(31\)	0.500815	−	0.241180i	0.0899490	−	0.0433172i	−0.388368	−	0.921504i	\(-0.626961\pi\)
							0.478317	+	0.878187i	\(0.341247\pi\)
\(37\)		−	9.70982i		−	1.59628i	−0.602469	−	0.798142i	\(-0.705816\pi\)
							0.602469	−	0.798142i	\(-0.294184\pi\)
\(41\)	0.854132	+	1.77362i	0.133393	+	0.276994i	0.956957	−	0.290230i	\(-0.0937318\pi\)
							−0.823564	+	0.567224i	\(0.808018\pi\)
\(43\)	−2.29150	−	6.14402i	−0.349451	−	0.936955i
							\(47\)	4.00745	+	3.19583i	0.584546	+	0.466160i	0.870558	−	0.492066i	\(-0.163758\pi\)
−0.286012	+	0.958226i	\(0.592330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.j.a.2.2	✓	72
3.2	odd	2	inner	129.2.j.a.2.11	yes	72
43.22	odd	14	inner	129.2.j.a.65.11	yes	72
129.65	even	14	inner	129.2.j.a.65.2	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.j.a.2.2	✓	72	1.1	even	1	trivial
129.2.j.a.2.11	yes	72	3.2	odd	2	inner
129.2.j.a.65.2	yes	72	129.65	even	14	inner
129.2.j.a.65.11	yes	72	43.22	odd	14	inner