Embedded newform 39.4.k.a.11.8

• Label: 39.4.k.a.11.8
• 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			11.8
Character	\(\chi\)	\(=\)	39.11
Dual form			39.4.k.a.32.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.933566 + 0.250148i) q^{2} +(-4.52105 - 2.56127i) q^{3} +(-6.11923 - 3.53294i) q^{4} +(11.3224 - 11.3224i) q^{5} +(-3.58000 - 3.52204i) q^{6} +(-3.58928 - 13.3954i) q^{7} +(-10.2963 - 10.2963i) q^{8} +(13.8798 + 23.1592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.933566	+	0.250148i	0.330065	+	0.0884407i	0.420046	−	0.907503i	\(-0.362014\pi\)
							−0.0899808	+	0.995944i	\(0.528681\pi\)
\(3\)	−4.52105	−	2.56127i	−0.870077	−	0.492916i
							\(5\)	11.3224	−	11.3224i	1.01270	−	1.01270i	0.0127864	−	0.999918i	\(-0.495930\pi\)
0.999918	−	0.0127864i	\(-0.00407014\pi\)
\(7\)	−3.58928	−	13.3954i	−0.193803	−	0.723283i	−0.992573	−	0.121647i	\(-0.961182\pi\)
							0.798770	−	0.601636i	\(-0.205484\pi\)
\(11\)	−13.6771	+	51.0438i	−0.374892	+	1.39912i	0.478611	+	0.878027i	\(0.341141\pi\)
							−0.853503	+	0.521089i	\(0.825526\pi\)
\(13\)	7.70792	−	46.2341i	0.164446	−	0.986386i
							\(17\)	39.8997	−	69.1083i	0.569241	−	0.985954i	−0.427401	−	0.904062i	\(-0.640571\pi\)
0.996641	−	0.0818915i	\(-0.0260961\pi\)
\(19\)	77.1147	−	20.6628i	0.931122	−	0.249494i	0.238789	−	0.971071i	\(-0.423250\pi\)
							0.692333	+	0.721578i	\(0.256583\pi\)
\(23\)	59.2117	+	102.558i	0.536804	+	0.929772i	0.999074	+	0.0430325i	\(0.0137019\pi\)
							−0.462270	+	0.886739i	\(0.652965\pi\)
\(29\)	84.1401	−	48.5783i	0.538773	−	0.311061i	−0.205808	−	0.978592i	\(-0.565982\pi\)
							0.744582	+	0.667531i	\(0.232649\pi\)
\(31\)	−86.1554	−	86.1554i	−0.499160	−	0.499160i	0.412016	−	0.911177i	\(-0.364825\pi\)
							−0.911177	+	0.412016i	\(0.864825\pi\)
\(37\)	24.6628	+	6.60839i	0.109582	+	0.0293625i	0.313193	−	0.949689i	\(-0.398601\pi\)
							−0.203611	+	0.979052i	\(0.565268\pi\)
\(41\)	−220.502	−	59.0834i	−0.839918	−	0.225055i	−0.186882	−	0.982382i	\(-0.559838\pi\)
							−0.653036	+	0.757327i	\(0.726505\pi\)
\(43\)	−34.1632	−	19.7241i	−0.121159	−	0.0699511i	0.438196	−	0.898880i	\(-0.355618\pi\)
							−0.559355	+	0.828928i	\(0.688951\pi\)
\(47\)	79.3929	+	79.3929i	0.246397	+	0.246397i	0.819490	−	0.573093i	\(-0.194257\pi\)
							−0.573093	+	0.819490i	\(0.694257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.11.8	yes	48
3.2	odd	2	inner	39.4.k.a.11.5	✓	48
13.6	odd	12	inner	39.4.k.a.32.5	yes	48
39.32	even	12	inner	39.4.k.a.32.8	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.11.5	✓	48	3.2	odd	2	inner
39.4.k.a.11.8	yes	48	1.1	even	1	trivial
39.4.k.a.32.5	yes	48	13.6	odd	12	inner
39.4.k.a.32.8	yes	48	39.32	even	12	inner