Embedded newform 39.4.k.a.2.4

• Label: 39.4.k.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	39.2
Dual form			39.4.k.a.20.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.614396 + 2.29296i) q^{2} +(-3.60382 + 3.74332i) q^{3} +(2.04803 + 1.18243i) q^{4} +(-2.27968 - 2.27968i) q^{5} +(-6.36910 - 10.5633i) q^{6} +(-25.9009 + 6.94012i) q^{7} +(-17.3981 + 17.3981i) q^{8} +(-1.02489 - 26.9805i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.614396	+	2.29296i	−0.217222	+	0.810683i	0.768151	+	0.640269i	\(0.221177\pi\)
							−0.985373	+	0.170414i	\(0.945490\pi\)
\(3\)	−3.60382	+	3.74332i	−0.693556	+	0.720402i
							\(5\)	−2.27968	−	2.27968i	−0.203901	−	0.203901i	0.597768	−	0.801669i	\(-0.296054\pi\)
−0.801669	+	0.597768i	\(0.796054\pi\)
\(7\)	−25.9009	+	6.94012i	−1.39852	+	0.374731i	−0.877812	−	0.479006i	\(-0.840997\pi\)
							−0.520704	+	0.853737i	\(0.674331\pi\)
\(11\)	59.0720	+	15.8283i	1.61917	+	0.433855i	0.950758	−	0.309935i	\(-0.100307\pi\)
							0.668412	+	0.743791i	\(0.266974\pi\)
\(13\)	19.7172	+	42.5233i	0.420660	+	0.907218i
							\(17\)	3.91452	−	6.78015i	0.0558477	−	0.0967311i	−0.836750	−	0.547585i	\(-0.815547\pi\)
0.892598	+	0.450854i	\(0.148881\pi\)
\(19\)	29.4876	+	110.049i	0.356048	+	1.32879i	0.879160	+	0.476526i	\(0.158104\pi\)
							−0.523112	+	0.852264i	\(0.675229\pi\)
\(23\)	5.49307	+	9.51427i	0.0497993	+	0.0862549i	0.889851	−	0.456252i	\(-0.150809\pi\)
							−0.840051	+	0.542507i	\(0.817475\pi\)
\(29\)	21.6113	−	12.4773i	0.138383	−	0.0798956i	−0.429210	−	0.903205i	\(-0.641208\pi\)
							0.567593	+	0.823309i	\(0.307875\pi\)
\(31\)	34.2500	−	34.2500i	0.198435	−	0.198435i	−0.600894	−	0.799329i	\(-0.705189\pi\)
							0.799329	+	0.600894i	\(0.205189\pi\)
\(37\)	35.8131	−	133.656i	0.159125	−	0.593864i	−0.839591	−	0.543219i	\(-0.817206\pi\)
							0.998717	−	0.0506454i	\(-0.0161278\pi\)
\(41\)	−53.8559	+	200.993i	−0.205143	+	0.765606i	0.784262	+	0.620429i	\(0.213042\pi\)
							−0.989406	+	0.145176i	\(0.953625\pi\)
\(43\)	−76.2601	−	44.0288i	−0.270455	−	0.156147i	0.358640	−	0.933476i	\(-0.383241\pi\)
							−0.629094	+	0.777329i	\(0.716574\pi\)
\(47\)	273.416	−	273.416i	0.848548	−	0.848548i	−0.141404	−	0.989952i	\(-0.545162\pi\)
							0.989952	+	0.141404i	\(0.0451615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.2.4	✓	48
3.2	odd	2	inner	39.4.k.a.2.9	yes	48
13.7	odd	12	inner	39.4.k.a.20.9	yes	48
39.20	even	12	inner	39.4.k.a.20.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.2.4	✓	48	1.1	even	1	trivial
39.4.k.a.2.9	yes	48	3.2	odd	2	inner
39.4.k.a.20.4	yes	48	39.20	even	12	inner
39.4.k.a.20.9	yes	48	13.7	odd	12	inner