Embedded newform 80.6.j.a.43.9

• Label: 80.6.j.a.43.9
• Level: $80$
• Weight: $6$
• Character: 80.43
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.9
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.15225 + 2.33545i) q^{2} -11.2637i q^{3} +(21.0913 - 24.0657i) q^{4} +(-52.1852 - 20.0427i) q^{5} +(26.3059 + 58.0336i) q^{6} +(-164.495 + 164.495i) q^{7} +(-52.4636 + 173.250i) q^{8} +116.128 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.15225	+	2.33545i	−0.910797	+	0.412853i
							\(3\)		−	11.2637i		−	0.722569i	−0.932456	−	0.361284i	\(-0.882338\pi\)
0.932456	−	0.361284i	\(-0.117662\pi\)
\(5\)	−52.1852	−	20.0427i	−0.933517	−	0.358534i
							\(7\)	−164.495	+	164.495i	−1.26884	+	1.26884i	−0.322159	+	0.946686i	\(0.604409\pi\)
−0.946686	+	0.322159i	\(0.895591\pi\)
\(11\)	62.0749	−	62.0749i	0.154680	−	0.154680i	−0.625524	−	0.780205i	\(-0.715115\pi\)
							0.780205	+	0.625524i	\(0.215115\pi\)
\(13\)	−208.945			−0.342905			−0.171452	−	0.985192i	\(-0.554846\pi\)
							−0.171452	+	0.985192i	\(0.554846\pi\)
\(17\)	707.991	−	707.991i	0.594163	−	0.594163i	−0.344590	−	0.938753i	\(-0.611982\pi\)
							0.938753	+	0.344590i	\(0.111982\pi\)
\(19\)	978.708	−	978.708i	0.621969	−	0.621969i	−0.324065	−	0.946035i	\(-0.605050\pi\)
							0.946035	+	0.324065i	\(0.105050\pi\)
\(23\)	3323.06	+	3323.06i	1.30984	+	1.30984i	0.921528	+	0.388313i	\(0.126942\pi\)
							0.388313	+	0.921528i	\(0.373058\pi\)
\(29\)	32.7520	+	32.7520i	0.00723173	+	0.00723173i	0.710713	−	0.703482i	\(-0.248372\pi\)
							−0.703482	+	0.710713i	\(0.748372\pi\)
\(31\)		−	467.990i		−	0.0874645i	−0.999043	−	0.0437323i	\(-0.986075\pi\)
							0.999043	−	0.0437323i	\(-0.0139249\pi\)
\(37\)	−2304.41			−0.276730			−0.138365	−	0.990381i	\(-0.544185\pi\)
							−0.138365	+	0.990381i	\(0.544185\pi\)
\(41\)			13403.7i			1.24527i	0.782512	+	0.622636i	\(0.213938\pi\)
							−0.782512	+	0.622636i	\(0.786062\pi\)
\(43\)	6558.74			0.540940			0.270470	−	0.962728i	\(-0.412821\pi\)
							0.270470	+	0.962728i	\(0.412821\pi\)
\(47\)	13878.8	+	13878.8i	0.916448	+	0.916448i	0.996769	−	0.0803211i	\(-0.0255946\pi\)
							−0.0803211	+	0.996769i	\(0.525595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.9	✓	116
5.2	odd	4		80.6.s.a.27.21	yes	116
16.3	odd	4		80.6.s.a.3.21	yes	116
80.67	even	4	inner	80.6.j.a.67.9	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.9	✓	116	1.1	even	1	trivial
80.6.j.a.67.9	yes	116	80.67	even	4	inner
80.6.s.a.3.21	yes	116	16.3	odd	4
80.6.s.a.27.21	yes	116	5.2	odd	4