Embedded newform 80.6.l.a.61.1

• Label: 80.6.l.a.61.1
• Level: $80$
• Weight: $6$
• Character: 80.61
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			61.1
Character	\(\chi\)	\(=\)	80.61
Dual form			80.6.l.a.21.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.65572 - 0.113075i) q^{2} +(-18.3942 - 18.3942i) q^{3} +(31.9744 + 1.27904i) q^{4} +(17.6777 - 17.6777i) q^{5} +(101.953 + 106.113i) q^{6} +47.3499i q^{7} +(-180.694 - 10.8494i) q^{8} +433.697i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.65572	−	0.113075i	−0.999800	−	0.0199890i
							\(3\)	−18.3942	−	18.3942i	−1.17999	−	1.17999i	−0.979746	−	0.200246i	\(-0.935826\pi\)
−0.200246	−	0.979746i	\(-0.564174\pi\)
\(5\)	17.6777	−	17.6777i	0.316228	−	0.316228i
							\(7\)			47.3499i			0.365236i	0.983184	+	0.182618i	\(0.0584572\pi\)
−0.983184	+	0.182618i	\(0.941543\pi\)
\(11\)	−285.234	+	285.234i	−0.710753	+	0.710753i	−0.966693	−	0.255940i	\(-0.917615\pi\)
							0.255940	+	0.966693i	\(0.417615\pi\)
\(13\)	301.631	+	301.631i	0.495013	+	0.495013i	0.909882	−	0.414868i	\(-0.136172\pi\)
							−0.414868	+	0.909882i	\(0.636172\pi\)
\(17\)	387.084			0.324850			0.162425	−	0.986721i	\(-0.448068\pi\)
							0.162425	+	0.986721i	\(0.448068\pi\)
\(19\)	1151.39	+	1151.39i	0.731709	+	0.731709i	0.970958	−	0.239250i	\(-0.0769014\pi\)
							−0.239250	+	0.970958i	\(0.576901\pi\)
\(23\)		−	4570.65i		−	1.80160i	−0.434234	−	0.900800i	\(-0.642981\pi\)
							0.434234	−	0.900800i	\(-0.357019\pi\)
\(29\)	5716.62	+	5716.62i	1.26225	+	1.26225i	0.950001	+	0.312246i	\(0.101081\pi\)
							0.312246	+	0.950001i	\(0.398919\pi\)
\(31\)	−5216.43			−0.974920			−0.487460	−	0.873145i	\(-0.662077\pi\)
							−0.487460	+	0.873145i	\(0.662077\pi\)
\(37\)	6376.99	−	6376.99i	0.765793	−	0.765793i	−0.211569	−	0.977363i	\(-0.567857\pi\)
							0.977363	+	0.211569i	\(0.0678575\pi\)
\(41\)		−	6877.46i		−	0.638952i	−0.947594	−	0.319476i	\(-0.896493\pi\)
							0.947594	−	0.319476i	\(-0.103507\pi\)
\(43\)	2857.60	−	2857.60i	0.235684	−	0.235684i	−0.579376	−	0.815060i	\(-0.696704\pi\)
							0.815060	+	0.579376i	\(0.196704\pi\)
\(47\)	−10988.9			−0.725622			−0.362811	−	0.931863i	\(-0.618183\pi\)
							−0.362811	+	0.931863i	\(0.618183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.61.1	yes	80
4.3	odd	2		320.6.l.a.81.37		80
16.5	even	4	inner	80.6.l.a.21.1	✓	80
16.11	odd	4		320.6.l.a.241.37		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.1	✓	80	16.5	even	4	inner
80.6.l.a.61.1	yes	80	1.1	even	1	trivial
320.6.l.a.81.37		80	4.3	odd	2
320.6.l.a.241.37		80	16.11	odd	4