Embedded newform 39.5.i.b.35.3

• Label: 39.5.i.b.35.3
• Level: $39$
• Weight: $5$
• Character: 39.35
• Analytic conductor: $4.031$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.03142856027\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			35.3
Character	\(\chi\)	\(=\)	39.35
Dual form			39.5.i.b.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.70393 - 3.29317i) q^{2} +(2.61666 + 8.61122i) q^{3} +(13.6899 + 23.7116i) q^{4} -26.9859i q^{5} +(13.4329 - 57.7349i) q^{6} +(-6.28578 - 10.8873i) q^{7} -74.9512i q^{8} +(-67.3062 + 45.0653i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{3} + 142 q^{4} + 122 q^{7} + 114 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.70393	−	3.29317i	−1.42598	−	0.823292i	−0.429183	−	0.903218i	\(-0.641198\pi\)
							−0.996801	+	0.0799259i	\(0.974532\pi\)
\(3\)	2.61666	+	8.61122i	0.290740	+	0.956802i
							\(5\)		−	26.9859i		−	1.07944i	−0.841846	−	0.539718i	\(-0.818531\pi\)
0.841846	−	0.539718i	\(-0.181469\pi\)
\(7\)	−6.28578	−	10.8873i	−0.128281	−	0.222190i	0.794730	−	0.606964i	\(-0.207613\pi\)
							−0.923011	+	0.384774i	\(0.874279\pi\)
\(11\)	−172.166	−	99.3999i	−1.42286	−	0.821486i	−0.426314	−	0.904575i	\(-0.640188\pi\)
							−0.996542	+	0.0830889i	\(0.973521\pi\)
\(13\)	−58.7418	−	158.463i	−0.347585	−	0.937649i
							\(17\)	−22.0167	+	12.7114i	−0.0761825	+	0.0439840i	−0.537607	−	0.843195i	\(-0.680672\pi\)
0.461425	+	0.887179i	\(0.347338\pi\)
\(19\)	−117.763	−	203.971i	−0.326213	−	0.565017i	0.655544	−	0.755157i	\(-0.272439\pi\)
							−0.981757	+	0.190140i	\(0.939106\pi\)
\(23\)	−503.541	−	290.720i	−0.951874	−	0.549565i	−0.0582113	−	0.998304i	\(-0.518540\pi\)
							−0.893663	+	0.448740i	\(0.851873\pi\)
\(29\)	1004.51	+	579.952i	1.19442	+	0.689599i	0.959306	−	0.282369i	\(-0.0911203\pi\)
							0.235114	+	0.971968i	\(0.424454\pi\)
\(31\)	527.121			0.548513			0.274257	−	0.961657i	\(-0.411568\pi\)
							0.274257	+	0.961657i	\(0.411568\pi\)
\(37\)	−1254.56	+	2172.97i	−0.916408	+	1.58727i	−0.111583	+	0.993755i	\(0.535592\pi\)
							−0.804826	+	0.593511i	\(0.797741\pi\)
\(41\)	1335.31	+	770.939i	0.794352	+	0.458619i	0.841492	−	0.540269i	\(-0.181678\pi\)
							−0.0471405	+	0.998888i	\(0.515011\pi\)
\(43\)	−1072.32	−	1857.32i	−0.579947	−	1.00450i	−0.995485	−	0.0949221i	\(-0.969740\pi\)
							0.415537	−	0.909576i	\(-0.363594\pi\)
\(47\)		−	2064.06i		−	0.934385i	−0.884156	−	0.467193i	\(-0.845265\pi\)
							0.884156	−	0.467193i	\(-0.154735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.i.b.35.3	yes	32
3.2	odd	2	inner	39.5.i.b.35.14	yes	32
13.3	even	3	inner	39.5.i.b.29.14	yes	32
39.29	odd	6	inner	39.5.i.b.29.3	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.i.b.29.3	✓	32	39.29	odd	6	inner
39.5.i.b.29.14	yes	32	13.3	even	3	inner
39.5.i.b.35.3	yes	32	1.1	even	1	trivial
39.5.i.b.35.14	yes	32	3.2	odd	2	inner