Embedded newform 80.6.s.a.3.16

• Label: 80.6.s.a.3.16
• Level: $80$
• Weight: $6$
• Character: 80.3
• 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.s (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			3.16
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.16

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.09704	+	3.90055i	−0.724260	+	0.689526i
							\(3\)	12.0492			0.772956			0.386478	−	0.922299i	\(-0.373692\pi\)
0.386478	+	0.922299i	\(0.373692\pi\)
\(5\)	−35.9351	+	42.8214i	−0.642826	+	0.766012i
							\(7\)	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	0.199060	−	0.979987i	\(-0.436211\pi\)
							−0.979987	+	0.199060i	\(0.936211\pi\)
\(13\)			952.447i			1.56308i	0.623852	+	0.781542i	\(0.285567\pi\)
							−0.623852	+	0.781542i	\(0.714433\pi\)
\(17\)	−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	\(-0.353231\pi\)
							−0.895568	+	0.444924i	\(0.853231\pi\)
\(23\)	−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	−0.158543	+	0.987352i	\(0.550680\pi\)
							−0.987352	+	0.158543i	\(0.949320\pi\)
\(31\)		−	9912.00i		−	1.85249i	−0.376916	−	0.926247i	\(-0.623015\pi\)
							0.376916	−	0.926247i	\(-0.376985\pi\)
\(37\)		−	10028.5i		−	1.20429i	−0.798385	−	0.602147i	\(-0.794312\pi\)
							0.798385	−	0.602147i	\(-0.205688\pi\)
\(41\)			12141.1i			1.12797i	0.825784	+	0.563987i	\(0.190733\pi\)
							−0.825784	+	0.563987i	\(0.809267\pi\)
\(43\)			5015.19i			0.413634i	0.978380	+	0.206817i	\(0.0663105\pi\)
							−0.978380	+	0.206817i	\(0.933689\pi\)
\(47\)	994.749	+	994.749i	0.0656854	+	0.0656854i	0.739186	−	0.673501i	\(-0.235210\pi\)
							−0.673501	+	0.739186i	\(0.735210\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.14	✓	116	16.11	odd	4
80.6.j.a.67.14	yes	116	5.2	odd	4
80.6.s.a.3.16	yes	116	1.1	even	1	trivial
80.6.s.a.27.16	yes	116	80.27	even	4	inner