Embedded newform 39.4.k.a.11.6

• Label: 39.4.k.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	39.11
Dual form			39.4.k.a.32.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.385245 - 0.103226i) q^{2} +(4.91747 - 1.67883i) q^{3} +(-6.79045 - 3.92047i) q^{4} +(9.27643 - 9.27643i) q^{5} +(-2.06773 + 0.139148i) q^{6} +(2.08748 + 7.79057i) q^{7} +(4.46744 + 4.46744i) q^{8} +(21.3631 - 16.5112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.385245	−	0.103226i	−0.136205	−	0.0364959i	0.190073	−	0.981770i	\(-0.439128\pi\)
							−0.326277	+	0.945274i	\(0.605794\pi\)
\(3\)	4.91747	−	1.67883i	0.946368	−	0.323090i
							\(5\)	9.27643	−	9.27643i	0.829709	−	0.829709i	−0.157767	−	0.987476i	\(-0.550430\pi\)
0.987476	+	0.157767i	\(0.0504296\pi\)
\(7\)	2.08748	+	7.79057i	0.112713	+	0.420651i	0.999106	−	0.0422831i	\(-0.0134631\pi\)
							−0.886392	+	0.462935i	\(0.846796\pi\)
\(11\)	4.51122	−	16.8361i	0.123653	−	0.461479i	−0.876135	−	0.482066i	\(-0.839887\pi\)
							0.999788	+	0.0205864i	\(0.00655332\pi\)
\(13\)	−7.38167	+	46.2873i	−0.157485	+	0.987521i
							\(17\)	−43.3605	+	75.1027i	−0.618616	+	1.07147i	0.371122	+	0.928584i	\(0.378973\pi\)
−0.989738	+	0.142891i	\(0.954360\pi\)
\(19\)	−81.1539	+	21.7451i	−0.979893	+	0.262562i	−0.713000	−	0.701164i	\(-0.752664\pi\)
							−0.266894	+	0.963726i	\(0.585997\pi\)
\(23\)	4.88572	+	8.46231i	0.0442932	+	0.0767180i	0.887322	−	0.461150i	\(-0.152563\pi\)
							−0.843029	+	0.537868i	\(0.819230\pi\)
\(29\)	218.975	−	126.425i	1.40216	−	0.809537i	0.407545	−	0.913185i	\(-0.366385\pi\)
							0.994614	+	0.103648i	\(0.0330515\pi\)
\(31\)	95.7551	+	95.7551i	0.554778	+	0.554778i	0.927816	−	0.373038i	\(-0.121684\pi\)
							−0.373038	+	0.927816i	\(0.621684\pi\)
\(37\)	−310.420	−	83.1767i	−1.37926	−	0.369572i	−0.508408	−	0.861116i	\(-0.669766\pi\)
							−0.870853	+	0.491544i	\(0.836433\pi\)
\(41\)	121.857	+	32.6516i	0.464169	+	0.124374i	0.483322	−	0.875443i	\(-0.339430\pi\)
							−0.0191526	+	0.999817i	\(0.506097\pi\)
\(43\)	126.080	+	72.7922i	0.447139	+	0.258156i	0.706621	−	0.707592i	\(-0.250219\pi\)
							−0.259482	+	0.965748i	\(0.583552\pi\)
\(47\)	−373.880	−	373.880i	−1.16034	−	1.16034i	−0.984401	−	0.175938i	\(-0.943704\pi\)
							−0.175938	−	0.984401i	\(-0.556296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.11.6	✓	48
3.2	odd	2	inner	39.4.k.a.11.7	yes	48
13.6	odd	12	inner	39.4.k.a.32.7	yes	48
39.32	even	12	inner	39.4.k.a.32.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.11.6	✓	48	1.1	even	1	trivial
39.4.k.a.11.7	yes	48	3.2	odd	2	inner
39.4.k.a.32.6	yes	48	39.32	even	12	inner
39.4.k.a.32.7	yes	48	13.6	odd	12	inner