Embedded newform 80.6.l.a.61.5

• Label: 80.6.l.a.61.5
• 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			61.5
Character	\(\chi\)	\(=\)	80.61
Dual form			80.6.l.a.21.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.42543 + 1.60146i) q^{2} +(3.86003 + 3.86003i) q^{3} +(26.8706 - 17.3773i) q^{4} +(-17.6777 + 17.6777i) q^{5} +(-27.1241 - 14.7606i) q^{6} -20.3706i q^{7} +(-117.956 + 137.312i) q^{8} -213.200i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.42543	+	1.60146i	−0.959090	+	0.283102i
							\(3\)	3.86003	+	3.86003i	0.247621	+	0.247621i	0.819994	−	0.572372i	\(-0.193977\pi\)
−0.572372	+	0.819994i	\(0.693977\pi\)
\(5\)	−17.6777	+	17.6777i	−0.316228	+	0.316228i
							\(7\)		−	20.3706i		−	0.157130i	−0.996909	−	0.0785649i	\(-0.974966\pi\)
0.996909	−	0.0785649i	\(-0.0250338\pi\)
\(11\)	−268.702	+	268.702i	−0.669558	+	0.669558i	−0.957614	−	0.288055i	\(-0.906991\pi\)
							0.288055	+	0.957614i	\(0.406991\pi\)
\(13\)	621.797	+	621.797i	1.02045	+	1.02045i	0.999787	+	0.0206604i	\(0.00657687\pi\)
							0.0206604	+	0.999787i	\(0.493423\pi\)
\(17\)	−2094.53			−1.75778			−0.878890	−	0.477025i	\(-0.841715\pi\)
							−0.878890	+	0.477025i	\(0.841715\pi\)
\(19\)	−370.318	−	370.318i	−0.235337	−	0.235337i	0.579579	−	0.814916i	\(-0.303217\pi\)
							−0.814916	+	0.579579i	\(0.803217\pi\)
\(23\)		−	3608.29i		−	1.42227i	−0.703055	−	0.711135i	\(-0.748181\pi\)
							0.703055	−	0.711135i	\(-0.251819\pi\)
\(29\)	−5957.69	−	5957.69i	−1.31548	−	1.31548i	−0.917316	−	0.398160i	\(-0.869649\pi\)
							−0.398160	−	0.917316i	\(-0.630351\pi\)
\(31\)	−10314.0			−1.92763			−0.963816	−	0.266569i	\(-0.914110\pi\)
							−0.963816	+	0.266569i	\(0.914110\pi\)
\(37\)	2830.29	−	2830.29i	0.339881	−	0.339881i	−0.516441	−	0.856322i	\(-0.672744\pi\)
							0.856322	+	0.516441i	\(0.172744\pi\)
\(41\)		−	2923.54i		−	0.271612i	−0.990735	−	0.135806i	\(-0.956638\pi\)
							0.990735	−	0.135806i	\(-0.0433624\pi\)
\(43\)	−7107.96	+	7107.96i	−0.586238	+	0.586238i	−0.936610	−	0.350372i	\(-0.886055\pi\)
							0.350372	+	0.936610i	\(0.386055\pi\)
\(47\)	−470.452			−0.0310650			−0.0155325	−	0.999879i	\(-0.504944\pi\)
							−0.0155325	+	0.999879i	\(0.504944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.61.5	yes	80
4.3	odd	2		320.6.l.a.81.17		80
16.5	even	4	inner	80.6.l.a.21.5	✓	80
16.11	odd	4		320.6.l.a.241.17		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.5	✓	80	16.5	even	4	inner
80.6.l.a.61.5	yes	80	1.1	even	1	trivial
320.6.l.a.81.17		80	4.3	odd	2
320.6.l.a.241.17		80	16.11	odd	4