Embedded newform 80.6.j.a.43.2

• Label: 80.6.j.a.43.2
• 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			43.2
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.55053 + 1.09161i) q^{2} -28.2900i q^{3} +(29.6168 - 12.1180i) q^{4} +(-13.1634 + 54.3298i) q^{5} +(30.8817 + 157.024i) q^{6} +(49.8932 - 49.8932i) q^{7} +(-151.161 + 99.5915i) q^{8} -557.323 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.55053	+	1.09161i	−0.981204	+	0.192971i
							\(3\)		−	28.2900i		−	1.81480i	−0.420264	−	0.907402i	\(-0.638063\pi\)
0.420264	−	0.907402i	\(-0.361937\pi\)
\(5\)	−13.1634	+	54.3298i	−0.235474	+	0.971881i
							\(7\)	49.8932	−	49.8932i	0.384854	−	0.384854i	−0.487993	−	0.872847i	\(-0.662271\pi\)
0.872847	+	0.487993i	\(0.162271\pi\)
\(11\)	−383.400	+	383.400i	−0.955368	+	0.955368i	−0.999046	−	0.0436772i	\(-0.986093\pi\)
							0.0436772	+	0.999046i	\(0.486093\pi\)
\(13\)	283.583			0.465396			0.232698	−	0.972549i	\(-0.425245\pi\)
							0.232698	+	0.972549i	\(0.425245\pi\)
\(17\)	−753.462	+	753.462i	−0.632323	+	0.632323i	−0.948650	−	0.316327i	\(-0.897550\pi\)
							0.316327	+	0.948650i	\(0.397550\pi\)
\(19\)	−43.2741	+	43.2741i	−0.0275007	+	0.0275007i	−0.720723	−	0.693223i	\(-0.756190\pi\)
							0.693223	+	0.720723i	\(0.256190\pi\)
\(23\)	2665.60	+	2665.60i	1.05069	+	1.05069i	0.998645	+	0.0520463i	\(0.0165743\pi\)
							0.0520463	+	0.998645i	\(0.483426\pi\)
\(29\)	−682.677	−	682.677i	−0.150737	−	0.150737i	0.627710	−	0.778447i	\(-0.283992\pi\)
							−0.778447	+	0.627710i	\(0.783992\pi\)
\(31\)			6935.11i			1.29613i	0.761584	+	0.648066i	\(0.224422\pi\)
							−0.761584	+	0.648066i	\(0.775578\pi\)
\(37\)	3033.16			0.364243			0.182121	−	0.983276i	\(-0.441704\pi\)
							0.182121	+	0.983276i	\(0.441704\pi\)
\(41\)		−	13429.9i		−	1.24771i	−0.781541	−	0.623854i	\(-0.785566\pi\)
							0.781541	−	0.623854i	\(-0.214434\pi\)
\(43\)	20416.8			1.68390			0.841949	−	0.539557i	\(-0.181408\pi\)
							0.841949	+	0.539557i	\(0.181408\pi\)
\(47\)	−13840.4	−	13840.4i	−0.913913	−	0.913913i	0.0826648	−	0.996577i	\(-0.473657\pi\)
							−0.996577	+	0.0826648i	\(0.973657\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.2	✓	116
5.2	odd	4		80.6.s.a.27.28	yes	116
16.3	odd	4		80.6.s.a.3.28	yes	116
80.67	even	4	inner	80.6.j.a.67.2	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.2	✓	116	1.1	even	1	trivial
80.6.j.a.67.2	yes	116	80.67	even	4	inner
80.6.s.a.3.28	yes	116	16.3	odd	4
80.6.s.a.27.28	yes	116	5.2	odd	4