Embedded newform 80.6.s.a.3.18

• Label: 80.6.s.a.3.18
• 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			3.18
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.31543 + 4.58344i) q^{2} -14.5939 q^{3} +(-10.0158 - 30.3922i) q^{4} +(-17.9275 - 52.9491i) q^{5} +(48.3851 - 66.8903i) q^{6} +(47.3215 - 47.3215i) q^{7} +(172.507 + 54.8563i) q^{8} -30.0178 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.31543	+	4.58344i	−0.586091	+	0.810245i
							\(3\)	−14.5939			−0.936200			−0.468100	−	0.883676i	\(-0.655061\pi\)
−0.468100	+	0.883676i	\(0.655061\pi\)
\(5\)	−17.9275	−	52.9491i	−0.320698	−	0.947182i
							\(7\)	47.3215	−	47.3215i	0.365018	−	0.365018i	−0.500639	−	0.865656i	\(-0.666902\pi\)
0.865656	+	0.500639i	\(0.166902\pi\)
\(11\)	28.1268	+	28.1268i	0.0700871	+	0.0700871i	0.741281	−	0.671194i	\(-0.234218\pi\)
							−0.671194	+	0.741281i	\(0.734218\pi\)
\(13\)		−	55.8021i		−	0.0915783i	−0.998951	−	0.0457891i	\(-0.985420\pi\)
							0.998951	−	0.0457891i	\(-0.0145802\pi\)
\(17\)	−92.8082	+	92.8082i	−0.0778868	+	0.0778868i	−0.744977	−	0.667090i	\(-0.767540\pi\)
							0.667090	+	0.744977i	\(0.267540\pi\)
\(19\)	−879.583	−	879.583i	−0.558975	−	0.558975i	0.370040	−	0.929016i	\(-0.379344\pi\)
							−0.929016	+	0.370040i	\(0.879344\pi\)
\(23\)	−128.064	−	128.064i	−0.0504786	−	0.0504786i	0.681417	−	0.731895i	\(-0.261364\pi\)
							−0.731895	+	0.681417i	\(0.761364\pi\)
\(29\)	−3399.85	+	3399.85i	−0.750696	+	0.750696i	−0.974609	−	0.223913i	\(-0.928117\pi\)
							0.223913	+	0.974609i	\(0.428117\pi\)
\(31\)			5968.27i			1.11543i	0.830031	+	0.557717i	\(0.188323\pi\)
							−0.830031	+	0.557717i	\(0.811677\pi\)
\(37\)			8959.01i			1.07586i	0.842990	+	0.537930i	\(0.180793\pi\)
							−0.842990	+	0.537930i	\(0.819207\pi\)
\(41\)			16429.3i			1.52636i	0.646183	+	0.763182i	\(0.276364\pi\)
							−0.646183	+	0.763182i	\(0.723636\pi\)
\(43\)		−	15387.3i		−	1.26909i	−0.772888	−	0.634543i	\(-0.781189\pi\)
							0.772888	−	0.634543i	\(-0.218811\pi\)
\(47\)	11254.2	+	11254.2i	0.743137	+	0.743137i	0.973180	−	0.230043i	\(-0.0738868\pi\)
							−0.230043	+	0.973180i	\(0.573887\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.18	yes	116
5.2	odd	4		80.6.j.a.67.12	yes	116
16.11	odd	4		80.6.j.a.43.12	✓	116
80.27	even	4	inner	80.6.s.a.27.18	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.12	✓	116	16.11	odd	4
80.6.j.a.67.12	yes	116	5.2	odd	4
80.6.s.a.3.18	yes	116	1.1	even	1	trivial
80.6.s.a.27.18	yes	116	80.27	even	4	inner