Embedded newform 39.4.k.a.2.12

• Label: 39.4.k.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	39.2
Dual form			39.4.k.a.20.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37323 - 5.12497i) q^{2} +(4.15460 + 3.12078i) q^{3} +(-17.4513 - 10.0755i) q^{4} +(-5.80773 - 5.80773i) q^{5} +(21.6992 - 17.0066i) q^{6} +(20.4169 - 5.47069i) q^{7} +(-45.5877 + 45.5877i) q^{8} +(7.52142 + 25.9312i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37323	−	5.12497i	0.485511	−	1.81195i	−0.0922395	−	0.995737i	\(-0.529403\pi\)
							0.577750	−	0.816214i	\(-0.303931\pi\)
\(3\)	4.15460	+	3.12078i	0.799553	+	0.600595i
							\(5\)	−5.80773	−	5.80773i	−0.519459	−	0.519459i	0.397949	−	0.917408i	\(-0.369722\pi\)
−0.917408	+	0.397949i	\(0.869722\pi\)
\(7\)	20.4169	−	5.47069i	1.10241	−	0.295390i	0.338664	−	0.940908i	\(-0.390025\pi\)
							0.763745	+	0.645518i	\(0.223358\pi\)
\(11\)	24.5957	+	6.59038i	0.674170	+	0.180643i	0.579632	−	0.814878i	\(-0.303196\pi\)
							0.0945374	+	0.995521i	\(0.469863\pi\)
\(13\)	7.53745	+	46.2622i	0.160809	+	0.986986i
							\(17\)	−12.2305	+	21.1839i	−0.174490	+	0.302226i	−0.939985	−	0.341216i	\(-0.889161\pi\)
0.765495	+	0.643442i	\(0.222494\pi\)
\(19\)	3.46687	+	12.9385i	0.0418607	+	0.156226i	0.983693	−	0.179858i	\(-0.0575639\pi\)
							−0.941832	+	0.336085i	\(0.890897\pi\)
\(23\)	−80.7600	−	139.880i	−0.732157	−	1.26813i	−0.955959	−	0.293499i	\(-0.905180\pi\)
							0.223802	−	0.974635i	\(-0.428153\pi\)
\(29\)	−170.568	+	98.4773i	−1.09219	+	0.630579i	−0.934160	−	0.356855i	\(-0.883849\pi\)
							−0.158034	+	0.987434i	\(0.550516\pi\)
\(31\)	−72.8478	+	72.8478i	−0.422060	+	0.422060i	−0.885912	−	0.463853i	\(-0.846467\pi\)
							0.463853	+	0.885912i	\(0.346467\pi\)
\(37\)	39.5244	−	147.507i	0.175615	−	0.655406i	−0.820831	−	0.571172i	\(-0.806489\pi\)
							0.996446	−	0.0842339i	\(-0.0268443\pi\)
\(41\)	−75.8144	+	282.943i	−0.288786	+	1.07776i	0.657242	+	0.753679i	\(0.271723\pi\)
							−0.946028	+	0.324084i	\(0.894944\pi\)
\(43\)	227.863	+	131.557i	0.808112	+	0.466564i	0.846300	−	0.532707i	\(-0.178825\pi\)
							−0.0381878	+	0.999271i	\(0.512158\pi\)
\(47\)	24.7856	−	24.7856i	0.0769223	−	0.0769223i	−0.667599	−	0.744521i	\(-0.732678\pi\)
							0.744521	+	0.667599i	\(0.232678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.2.12	yes	48
3.2	odd	2	inner	39.4.k.a.2.1	✓	48
13.7	odd	12	inner	39.4.k.a.20.1	yes	48
39.20	even	12	inner	39.4.k.a.20.12	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.2.1	✓	48	3.2	odd	2	inner
39.4.k.a.2.12	yes	48	1.1	even	1	trivial
39.4.k.a.20.1	yes	48	13.7	odd	12	inner
39.4.k.a.20.12	yes	48	39.20	even	12	inner