Embedded newform 39.4.f.b.5.7

• Label: 39.4.f.b.5.7
• Level: $39$
• Weight: $4$
• Character: 39.5
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	\( x^{20} + 1316x^{16} + 520390x^{12} + 64668772x^{8} + 2536036097x^{4} + 8509693504 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.7
Root			\(2.05488 - 2.05488i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.5
Dual form			39.4.f.b.8.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.05488 + 2.05488i) q^{2} +(0.159445 - 5.19371i) q^{3} +0.445059i q^{4} +(14.0419 + 14.0419i) q^{5} +(11.0001 - 10.3448i) q^{6} +(-5.62483 - 5.62483i) q^{7} +(15.5245 - 15.5245i) q^{8} +(-26.9492 - 1.65622i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.05488	+	2.05488i	0.726510	+	0.726510i	0.969923	−	0.243413i	\(-0.0782671\pi\)
							−0.243413	+	0.969923i	\(0.578267\pi\)
\(3\)	0.159445	−	5.19371i	0.0306852	−	0.999529i
							\(5\)	14.0419	+	14.0419i	1.25594	+	1.25594i	0.953012	+	0.302933i	\(0.0979658\pi\)
0.302933	+	0.953012i	\(0.402034\pi\)
\(7\)	−5.62483	−	5.62483i	−0.303712	−	0.303712i	0.538752	−	0.842464i	\(-0.318896\pi\)
							−0.842464	+	0.538752i	\(0.818896\pi\)
\(11\)	−25.0846	+	25.0846i	−0.687571	+	0.687571i	−0.961695	−	0.274123i	\(-0.911612\pi\)
							0.274123	+	0.961695i	\(0.411612\pi\)
\(13\)	−46.8481	−	1.50111i	−0.999487	−	0.0320257i
							\(17\)	−82.2495			−1.17344			−0.586718	−	0.809791i	\(-0.699580\pi\)
−0.586718	+	0.809791i	\(0.699580\pi\)
\(19\)	29.4522	−	29.4522i	0.355621	−	0.355621i	−0.506575	−	0.862196i	\(-0.669089\pi\)
							0.862196	+	0.506575i	\(0.169089\pi\)
\(23\)	107.699			0.976385			0.488193	−	0.872736i	\(-0.337656\pi\)
							0.488193	+	0.872736i	\(0.337656\pi\)
\(29\)			29.9509i			0.191784i	0.995392	+	0.0958920i	\(0.0305704\pi\)
							−0.995392	+	0.0958920i	\(0.969430\pi\)
\(31\)	62.0870	−	62.0870i	0.359715	−	0.359715i	−0.503993	−	0.863708i	\(-0.668136\pi\)
							0.863708	+	0.503993i	\(0.168136\pi\)
\(37\)	−132.465	−	132.465i	−0.588572	−	0.588572i	0.348673	−	0.937244i	\(-0.386632\pi\)
							−0.937244	+	0.348673i	\(0.886632\pi\)
\(41\)	238.610	+	238.610i	0.908895	+	0.908895i	0.996183	−	0.0872882i	\(-0.0278201\pi\)
							−0.0872882	+	0.996183i	\(0.527820\pi\)
\(43\)		−	140.796i		−	0.499328i	−0.968333	−	0.249664i	\(-0.919680\pi\)
							0.968333	−	0.249664i	\(-0.0803202\pi\)
\(47\)	172.151	−	172.151i	0.534273	−	0.534273i	−0.387568	−	0.921841i	\(-0.626685\pi\)
							0.921841	+	0.387568i	\(0.126685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.f.b.5.7	yes	20
3.2	odd	2	inner	39.4.f.b.5.4	✓	20
13.8	odd	4	inner	39.4.f.b.8.4	yes	20
39.8	even	4	inner	39.4.f.b.8.7	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.f.b.5.4	✓	20	3.2	odd	2	inner
39.4.f.b.5.7	yes	20	1.1	even	1	trivial
39.4.f.b.8.4	yes	20	13.8	odd	4	inner
39.4.f.b.8.7	yes	20	39.8	even	4	inner