Embedded newform 80.6.j.a.43.14

• Label: 80.6.j.a.43.14
• 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			43.14
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.90055 + 4.09704i) q^{2} +12.0492i q^{3} +(-1.57141 - 31.9614i) q^{4} +(-42.8214 - 35.9351i) q^{5} +(-49.3660 - 46.9985i) q^{6} +(63.2042 - 63.2042i) q^{7} +(137.076 + 118.229i) q^{8} +97.8169 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.90055	+	4.09704i	−0.689526	+	0.724260i
							\(3\)			12.0492i			0.772956i	0.922299	+	0.386478i	\(0.126308\pi\)
−0.922299	+	0.386478i	\(0.873692\pi\)
\(5\)	−42.8214	−	35.9351i	−0.766012	−	0.642826i
							\(7\)	63.2042	−	63.2042i	0.487530	−	0.487530i	−0.419996	−	0.907526i	\(-0.637969\pi\)
0.907526	+	0.419996i	\(0.137969\pi\)
\(11\)	−313.395	+	313.395i	−0.780927	+	0.780927i	−0.979987	−	0.199060i	\(-0.936211\pi\)
							0.199060	+	0.979987i	\(0.436211\pi\)
\(13\)	952.447			1.56308			0.781542	−	0.623852i	\(-0.214433\pi\)
							0.781542	+	0.623852i	\(0.214433\pi\)
\(17\)	−554.160	+	554.160i	−0.465064	+	0.465064i	−0.900311	−	0.435247i	\(-0.856661\pi\)
							0.435247	+	0.900311i	\(0.356661\pi\)
\(19\)	709.118	−	709.118i	0.450645	−	0.450645i	−0.444924	−	0.895568i	\(-0.646769\pi\)
							0.895568	+	0.444924i	\(0.146769\pi\)
\(23\)	−1828.89	−	1828.89i	−0.720890	−	0.720890i	0.247896	−	0.968787i	\(-0.420261\pi\)
							−0.968787	+	0.247896i	\(0.920261\pi\)
\(29\)	5189.67	+	5189.67i	1.14589	+	1.14589i	0.987352	+	0.158543i	\(0.0506796\pi\)
							0.158543	+	0.987352i	\(0.449320\pi\)
\(31\)			9912.00i			1.85249i	0.376916	+	0.926247i	\(0.376985\pi\)
							−0.376916	+	0.926247i	\(0.623015\pi\)
\(37\)	10028.5			1.20429			0.602147	−	0.798385i	\(-0.294312\pi\)
							0.602147	+	0.798385i	\(0.294312\pi\)
\(41\)		−	12141.1i		−	1.12797i	−0.825784	−	0.563987i	\(-0.809267\pi\)
							0.825784	−	0.563987i	\(-0.190733\pi\)
\(43\)	5015.19			0.413634			0.206817	−	0.978380i	\(-0.433689\pi\)
							0.206817	+	0.978380i	\(0.433689\pi\)
\(47\)	−994.749	−	994.749i	−0.0656854	−	0.0656854i	0.673501	−	0.739186i	\(-0.264790\pi\)
							−0.739186	+	0.673501i	\(0.764790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.14	✓	116
5.2	odd	4		80.6.s.a.27.16	yes	116
16.3	odd	4		80.6.s.a.3.16	yes	116
80.67	even	4	inner	80.6.j.a.67.14	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.14	✓	116	1.1	even	1	trivial
80.6.j.a.67.14	yes	116	80.67	even	4	inner
80.6.s.a.3.16	yes	116	16.3	odd	4
80.6.s.a.27.16	yes	116	5.2	odd	4