Embedded newform 39.5.i.b.35.4

• Label: 39.5.i.b.35.4
• 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			35.4
Character	\(\chi\)	\(=\)	39.35
Dual form			39.5.i.b.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.18280 - 2.41494i) q^{2} +(-2.70769 - 8.58303i) q^{3} +(3.66386 + 6.34600i) q^{4} -24.8103i q^{5} +(-9.40176 + 42.4400i) q^{6} +(-4.36411 - 7.55885i) q^{7} +41.8860i q^{8} +(-66.3368 + 46.4804i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{3} + 142 q^{4} + 122 q^{7} + 114 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.18280	−	2.41494i	−1.04570	−	0.603735i	−0.124257	−	0.992250i	\(-0.539655\pi\)
							−0.921442	+	0.388515i	\(0.872988\pi\)
\(3\)	−2.70769	−	8.58303i	−0.300855	−	0.953670i
							\(5\)		−	24.8103i		−	0.992412i	−0.868205	−	0.496206i	\(-0.834726\pi\)
0.868205	−	0.496206i	\(-0.165274\pi\)
\(7\)	−4.36411	−	7.55885i	−0.0890634	−	0.154262i	0.818052	−	0.575144i	\(-0.195054\pi\)
							−0.907115	+	0.420882i	\(0.861721\pi\)
\(11\)	44.8276	+	25.8812i	0.370476	+	0.213894i	0.673666	−	0.739036i	\(-0.264718\pi\)
							−0.303190	+	0.952930i	\(0.598052\pi\)
\(13\)	−128.880	+	109.320i	−0.762606	+	0.646863i
							\(17\)	135.624	−	78.3024i	0.469286	−	0.270943i	−0.246655	−	0.969103i	\(-0.579331\pi\)
0.715941	+	0.698161i	\(0.245998\pi\)
\(19\)	−237.502	−	411.365i	−0.657899	−	1.13951i	−0.981159	−	0.193204i	\(-0.938112\pi\)
							0.323259	−	0.946310i	\(-0.395221\pi\)
\(23\)	36.3588	+	20.9918i	0.0687313	+	0.0396820i	0.533972	−	0.845502i	\(-0.320699\pi\)
							−0.465241	+	0.885184i	\(0.654032\pi\)
\(29\)	−846.269	−	488.594i	−1.00626	−	0.580967i	−0.0961690	−	0.995365i	\(-0.530659\pi\)
							−0.910096	+	0.414398i	\(0.863992\pi\)
\(31\)	−1487.51			−1.54788			−0.773939	−	0.633261i	\(-0.781716\pi\)
							−0.773939	+	0.633261i	\(0.781716\pi\)
\(37\)	−179.074	+	310.164i	−0.130806	+	0.226563i	−0.923988	−	0.382423i	\(-0.875090\pi\)
							0.793181	+	0.608985i	\(0.208423\pi\)
\(41\)	−2429.58	−	1402.72i	−1.44532	−	0.834456i	−0.447124	−	0.894472i	\(-0.647552\pi\)
							−0.998197	+	0.0600157i	\(0.980885\pi\)
\(43\)	1215.16	+	2104.72i	0.657199	+	1.13830i	0.981338	+	0.192292i	\(0.0615921\pi\)
							−0.324139	+	0.946009i	\(0.605075\pi\)
\(47\)		−	172.507i		−	0.0780928i	−0.999237	−	0.0390464i	\(-0.987568\pi\)
							0.999237	−	0.0390464i	\(-0.0124320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.i.b.35.4	yes	32
3.2	odd	2	inner	39.5.i.b.35.13	yes	32
13.3	even	3	inner	39.5.i.b.29.13	yes	32
39.29	odd	6	inner	39.5.i.b.29.4	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.i.b.29.4	✓	32	39.29	odd	6	inner
39.5.i.b.29.13	yes	32	13.3	even	3	inner
39.5.i.b.35.4	yes	32	1.1	even	1	trivial
39.5.i.b.35.13	yes	32	3.2	odd	2	inner