Embedded newform 39.4.k.a.11.4

• Label: 39.4.k.a.11.4
• Level: $39$
• Weight: $4$
• Character: 39.11
• 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			11.4
Character	\(\chi\)	\(=\)	39.11
Dual form			39.4.k.a.32.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.98884 - 0.800858i) q^{2} +(3.86434 + 3.47374i) q^{3} +(1.36361 + 0.787281i) q^{4} +(-6.51442 + 6.51442i) q^{5} +(-8.76795 - 13.4773i) q^{6} +(4.63030 + 17.2805i) q^{7} +(14.0588 + 14.0588i) q^{8} +(2.86629 + 26.8474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.98884	−	0.800858i	−1.05672	−	0.283146i	−0.311691	−	0.950183i	\(-0.600896\pi\)
							−0.745024	+	0.667037i	\(0.767562\pi\)
\(3\)	3.86434	+	3.47374i	0.743693	+	0.668521i
							\(5\)	−6.51442	+	6.51442i	−0.582667	+	0.582667i	−0.935635	−	0.352968i	\(-0.885172\pi\)
0.352968	+	0.935635i	\(0.385172\pi\)
\(7\)	4.63030	+	17.2805i	0.250013	+	0.933059i	0.970797	+	0.239901i	\(0.0771149\pi\)
							−0.720785	+	0.693159i	\(0.756218\pi\)
\(11\)	−10.6618	+	39.7905i	−0.292242	+	1.09066i	0.651141	+	0.758957i	\(0.274291\pi\)
							−0.943383	+	0.331705i	\(0.892376\pi\)
\(13\)	−1.70529	−	46.8411i	−0.0363817	−	0.999338i
							\(17\)	18.9370	−	32.7999i	0.270171	−	0.467949i	−0.698735	−	0.715381i	\(-0.746253\pi\)
0.968905	+	0.247432i	\(0.0795865\pi\)
\(19\)	11.2790	−	3.02221i	0.136189	−	0.0364917i	−0.190081	−	0.981768i	\(-0.560875\pi\)
							0.326269	+	0.945277i	\(0.394208\pi\)
\(23\)	−36.7444	−	63.6432i	−0.333119	−	0.576979i	0.650003	−	0.759932i	\(-0.274768\pi\)
							−0.983122	+	0.182953i	\(0.941434\pi\)
\(29\)	261.382	−	150.909i	1.67370	−	0.966312i	0.708165	−	0.706047i	\(-0.249523\pi\)
							0.965537	−	0.260265i	\(-0.0838099\pi\)
\(31\)	144.820	+	144.820i	0.839045	+	0.839045i	0.988733	−	0.149688i	\(-0.0478270\pi\)
							−0.149688	+	0.988733i	\(0.547827\pi\)
\(37\)	−53.9159	−	14.4467i	−0.239560	−	0.0641898i	0.137041	−	0.990565i	\(-0.456241\pi\)
							−0.376601	+	0.926375i	\(0.622907\pi\)
\(41\)	−297.206	−	79.6362i	−1.13209	−	0.303343i	−0.356325	−	0.934362i	\(-0.615970\pi\)
							−0.775768	+	0.631019i	\(0.782637\pi\)
\(43\)	136.516	+	78.8173i	0.484150	+	0.279524i	0.722144	−	0.691743i	\(-0.243157\pi\)
							−0.237995	+	0.971266i	\(0.576490\pi\)
\(47\)	260.789	+	260.789i	0.809362	+	0.809362i	0.984537	−	0.175175i	\(-0.0560491\pi\)
							−0.175175	+	0.984537i	\(0.556049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.11.4	✓	48
3.2	odd	2	inner	39.4.k.a.11.9	yes	48
13.6	odd	12	inner	39.4.k.a.32.9	yes	48
39.32	even	12	inner	39.4.k.a.32.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.11.4	✓	48	1.1	even	1	trivial
39.4.k.a.11.9	yes	48	3.2	odd	2	inner
39.4.k.a.32.4	yes	48	39.32	even	12	inner
39.4.k.a.32.9	yes	48	13.6	odd	12	inner