Embedded newform 39.4.j.c.4.1

• Label: 39.4.j.c.4.1
• Level: $39$
• Weight: $4$
• Character: 39.4
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial:	\( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.1
Root			\(-5.36472i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.4
Dual form			39.4.j.c.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.64599 - 2.68236i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(10.3901 + 17.9962i) q^{4} -2.69631i q^{5} +(13.9380 - 8.04709i) q^{6} +(13.1657 - 7.60123i) q^{7} -68.5626i q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 15 q^{3} + 30 q^{4} + 30 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.64599	−	2.68236i	−1.64260	−	0.948358i	−0.979903	−	0.199475i	\(-0.936076\pi\)
							−0.662702	−	0.748883i	\(-0.730590\pi\)
\(3\)	−1.50000	+	2.59808i	−0.288675	+	0.500000i
							\(5\)		−	2.69631i		−	0.241165i	−0.992703	−	0.120583i	\(-0.961524\pi\)
0.992703	−	0.120583i	\(-0.0384763\pi\)
\(7\)	13.1657	−	7.60123i	0.710882	−	0.410428i	−0.100505	−	0.994937i	\(-0.532046\pi\)
							0.811388	+	0.584509i	\(0.198713\pi\)
\(11\)	57.9240	+	33.4424i	1.58770	+	0.916661i	0.993685	+	0.112209i	\(0.0357927\pi\)
							0.594018	+	0.804451i	\(0.297541\pi\)
\(13\)	46.8650	−	0.818689i	0.999847	−	0.0174664i
							\(17\)	2.08177	+	3.60573i	0.0297002	+	0.0514422i	0.880493	−	0.474058i	\(-0.157211\pi\)
−0.850793	+	0.525501i	\(0.823878\pi\)
\(19\)	−22.5903	+	13.0425i	−0.272766	+	0.157482i	−0.630144	−	0.776478i	\(-0.717004\pi\)
							0.357378	+	0.933960i	\(0.383671\pi\)
\(23\)	23.6621	−	40.9839i	0.214517	−	0.371554i	−0.738606	−	0.674137i	\(-0.764516\pi\)
							0.953123	+	0.302583i	\(0.0978490\pi\)
\(29\)	−128.503	+	222.575i	−0.822845	+	1.42521i	0.0807106	+	0.996738i	\(0.474281\pi\)
							−0.903555	+	0.428471i	\(0.859052\pi\)
\(31\)		−	206.242i		−	1.19491i	−0.801903	−	0.597455i	\(-0.796179\pi\)
							0.801903	−	0.597455i	\(-0.203821\pi\)
\(37\)	−152.149	−	87.8430i	−0.676029	−	0.390305i	0.122328	−	0.992490i	\(-0.460964\pi\)
							−0.798357	+	0.602184i	\(0.794297\pi\)
\(41\)	135.501	+	78.2313i	0.516137	+	0.297992i	0.735353	−	0.677684i	\(-0.237016\pi\)
							−0.219216	+	0.975676i	\(0.570350\pi\)
\(43\)	25.9922	+	45.0199i	0.0921809	+	0.159662i	0.908429	−	0.418040i	\(-0.137283\pi\)
							−0.816248	+	0.577702i	\(0.803950\pi\)
\(47\)		−	354.222i		−	1.09933i	−0.835384	−	0.549666i	\(-0.814755\pi\)
							0.835384	−	0.549666i	\(-0.185245\pi\)