Embedded newform 39.4.k.a.2.11

• Label: 39.4.k.a.2.11
• Level: $39$
• Weight: $4$
• Character: 39.2
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	39.2
Dual form			39.4.k.a.20.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13863 - 4.24944i) q^{2} +(-1.01733 - 5.09559i) q^{3} +(-9.83302 - 5.67709i) q^{4} +(8.66436 + 8.66436i) q^{5} +(-22.8117 - 1.47892i) q^{6} +(-1.76245 + 0.472246i) q^{7} +(-10.4342 + 10.4342i) q^{8} +(-24.9301 + 10.3678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13863	−	4.24944i	0.402568	−	1.50240i	−0.405931	−	0.913904i	\(-0.633053\pi\)
							0.808498	−	0.588499i	\(-0.200281\pi\)
\(3\)	−1.01733	−	5.09559i	−0.195786	−	0.980647i
							\(5\)	8.66436	+	8.66436i	0.774964	+	0.774964i	0.978970	−	0.204006i	\(-0.0653962\pi\)
−0.204006	+	0.978970i	\(0.565396\pi\)
\(7\)	−1.76245	+	0.472246i	−0.0951632	+	0.0254989i	−0.306086	−	0.952004i	\(-0.599020\pi\)
							0.210923	+	0.977503i	\(0.432353\pi\)
\(11\)	39.3062	+	10.5321i	1.07739	+	0.288685i	0.753525	−	0.657419i	\(-0.228352\pi\)
							0.323862	+	0.946104i	\(0.395019\pi\)
\(13\)	−21.4998	−	41.6504i	−0.458690	−	0.888596i
							\(17\)	25.7738	−	44.6415i	0.367709	−	0.636891i	−0.621498	−	0.783416i	\(-0.713475\pi\)
0.989207	+	0.146525i	\(0.0468088\pi\)
\(19\)	41.2000	+	153.760i	0.497469	+	1.85658i	0.515736	+	0.856748i	\(0.327519\pi\)
							−0.0182666	+	0.999833i	\(0.505815\pi\)
\(23\)	48.8713	+	84.6476i	0.443060	+	0.767402i	0.997915	−	0.0645453i	\(-0.0205597\pi\)
							−0.554855	+	0.831947i	\(0.687226\pi\)
\(29\)	−114.571	+	66.1476i	−0.733631	+	0.423562i	−0.819749	−	0.572723i	\(-0.805887\pi\)
							0.0861180	+	0.996285i	\(0.472554\pi\)
\(31\)	−72.6391	+	72.6391i	−0.420850	+	0.420850i	−0.885496	−	0.464646i	\(-0.846182\pi\)
							0.464646	+	0.885496i	\(0.346182\pi\)
\(37\)	23.4241	−	87.4201i	0.104079	−	0.388426i	−0.894160	−	0.447747i	\(-0.852227\pi\)
							0.998239	+	0.0593203i	\(0.0188933\pi\)
\(41\)	−15.6977	+	58.5847i	−0.0597945	+	0.223156i	−0.989357	−	0.145508i	\(-0.953518\pi\)
							0.929563	+	0.368665i	\(0.120185\pi\)
\(43\)	−94.0760	−	54.3148i	−0.333638	−	0.192626i	0.323817	−	0.946120i	\(-0.395034\pi\)
							−0.657455	+	0.753494i	\(0.728367\pi\)
\(47\)	313.954	−	313.954i	0.974359	−	0.974359i	−0.0253206	−	0.999679i	\(-0.508061\pi\)
							0.999679	+	0.0253206i	\(0.00806066\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.2.11	yes	48
3.2	odd	2	inner	39.4.k.a.2.2	✓	48
13.7	odd	12	inner	39.4.k.a.20.2	yes	48
39.20	even	12	inner	39.4.k.a.20.11	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.2.2	✓	48	3.2	odd	2	inner
39.4.k.a.2.11	yes	48	1.1	even	1	trivial
39.4.k.a.20.2	yes	48	13.7	odd	12	inner
39.4.k.a.20.11	yes	48	39.20	even	12	inner