Embedded newform 39.4.k.a.11.2

• Label: 39.4.k.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	39.11
Dual form			39.4.k.a.32.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.88819 - 1.04184i) q^{2} +(-4.68804 - 2.24104i) q^{3} +(7.10439 + 4.10172i) q^{4} +(0.168873 - 0.168873i) q^{5} +(15.8932 + 13.5978i) q^{6} +(5.26279 + 19.6410i) q^{7} +(-0.579067 - 0.579067i) q^{8} +(16.9555 + 21.0122i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.88819	−	1.04184i	−1.37468	−	0.368345i	−0.505496	−	0.862829i	\(-0.668690\pi\)
							−0.869187	+	0.494484i	\(0.835357\pi\)
\(3\)	−4.68804	−	2.24104i	−0.902214	−	0.431289i
							\(5\)	0.168873	−	0.168873i	0.0151044	−	0.0151044i	−0.699514	−	0.714619i	\(-0.746600\pi\)
0.714619	+	0.699514i	\(0.246600\pi\)
\(7\)	5.26279	+	19.6410i	0.284164	+	1.06051i	0.949448	+	0.313924i	\(0.101644\pi\)
							−0.665284	+	0.746590i	\(0.731690\pi\)
\(11\)	3.80862	−	14.2140i	0.104395	−	0.389607i	−0.893881	−	0.448304i	\(-0.852028\pi\)
							0.998276	+	0.0586975i	\(0.0186947\pi\)
\(13\)	−27.1218	+	38.2284i	−0.578634	+	0.815587i
							\(17\)	−40.6399	+	70.3904i	−0.579802	+	1.00425i	0.415700	+	0.909502i	\(0.363537\pi\)
−0.995502	+	0.0947440i	\(0.969797\pi\)
\(19\)	41.2307	−	11.0477i	0.497841	−	0.133396i	−0.00115701	−	0.999999i	\(-0.500368\pi\)
							0.498998	+	0.866603i	\(0.333702\pi\)
\(23\)	65.0715	+	112.707i	0.589928	+	1.02179i	0.994241	+	0.107165i	\(0.0341772\pi\)
							−0.404313	+	0.914621i	\(0.632489\pi\)
\(29\)	−138.170	+	79.7727i	−0.884744	+	0.510807i	−0.872220	−	0.489114i	\(-0.837320\pi\)
							−0.0125246	+	0.999922i	\(0.503987\pi\)
\(31\)	−9.94310	−	9.94310i	−0.0576076	−	0.0576076i	0.677716	−	0.735324i	\(-0.262970\pi\)
							−0.735324	+	0.677716i	\(0.762970\pi\)
\(37\)	315.375	+	84.5044i	1.40128	+	0.375471i	0.878804	−	0.477183i	\(-0.158342\pi\)
							0.522475	+	0.852655i	\(0.325009\pi\)
\(41\)	−53.2336	−	14.2639i	−0.202773	−	0.0543328i	0.156003	−	0.987757i	\(-0.450139\pi\)
							−0.358776	+	0.933424i	\(0.616806\pi\)
\(43\)	−434.660	−	250.951i	−1.54151	−	0.889992i	−0.998744	−	0.0501088i	\(-0.984043\pi\)
							−0.542767	−	0.839883i	\(-0.682623\pi\)
\(47\)	−325.007	−	325.007i	−1.00866	−	1.00866i	−0.999962	−	0.00870149i	\(-0.997230\pi\)
							−0.00870149	−	0.999962i	\(-0.502770\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.k.a.11.2	✓	48
3.2	odd	2	inner	39.4.k.a.11.11	yes	48
13.6	odd	12	inner	39.4.k.a.32.11	yes	48
39.32	even	12	inner	39.4.k.a.32.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.k.a.11.2	✓	48	1.1	even	1	trivial
39.4.k.a.11.11	yes	48	3.2	odd	2	inner
39.4.k.a.32.2	yes	48	39.32	even	12	inner
39.4.k.a.32.11	yes	48	13.6	odd	12	inner