Embedded newform 80.6.s.a.3.6

• Label: 80.6.s.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.46635 + 1.45570i) q^{2} -23.1807 q^{3} +(27.7619 - 15.9147i) q^{4} +(47.1776 - 29.9879i) q^{5} +(126.714 - 33.7441i) q^{6} +(-125.489 + 125.489i) q^{7} +(-128.589 + 127.408i) q^{8} +294.343 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.46635	+	1.45570i	−0.966322	+	0.257334i
							\(3\)	−23.1807			−1.48704			−0.743520	−	0.668714i	\(-0.766845\pi\)
−0.743520	+	0.668714i	\(0.766845\pi\)
\(5\)	47.1776	−	29.9879i	0.843938	−	0.536440i
							\(7\)	−125.489	+	125.489i	−0.967967	+	0.967967i	−0.999503	−	0.0315355i	\(-0.989960\pi\)
0.0315355	+	0.999503i	\(0.489960\pi\)
\(11\)	169.689	+	169.689i	0.422836	+	0.422836i	0.886179	−	0.463343i	\(-0.153350\pi\)
							−0.463343	+	0.886179i	\(0.653350\pi\)
\(13\)			898.070i			1.47385i	0.675977	+	0.736923i	\(0.263722\pi\)
							−0.675977	+	0.736923i	\(0.736278\pi\)
\(17\)	934.475	−	934.475i	0.784233	−	0.784233i	−0.196309	−	0.980542i	\(-0.562895\pi\)
							0.980542	+	0.196309i	\(0.0628955\pi\)
\(19\)	−1371.91	−	1371.91i	−0.871852	−	0.871852i	0.120822	−	0.992674i	\(-0.461447\pi\)
							−0.992674	+	0.120822i	\(0.961447\pi\)
\(23\)	−1148.91	−	1148.91i	−0.452864	−	0.452864i	0.443440	−	0.896304i	\(-0.353758\pi\)
							−0.896304	+	0.443440i	\(0.853758\pi\)
\(29\)	−3403.54	+	3403.54i	−0.751512	+	0.751512i	−0.974761	−	0.223250i	\(-0.928333\pi\)
							0.223250	+	0.974761i	\(0.428333\pi\)
\(31\)			1049.38i			0.196123i	0.995180	+	0.0980617i	\(0.0312643\pi\)
							−0.995180	+	0.0980617i	\(0.968736\pi\)
\(37\)		−	15735.4i		−	1.88962i	−0.327624	−	0.944808i	\(-0.606248\pi\)
							0.327624	−	0.944808i	\(-0.393752\pi\)
\(41\)		−	4036.97i		−	0.375056i	−0.982259	−	0.187528i	\(-0.939952\pi\)
							0.982259	−	0.187528i	\(-0.0600475\pi\)
\(43\)		−	1798.08i		−	0.148299i	−0.997247	−	0.0741493i	\(-0.976376\pi\)
							0.997247	−	0.0741493i	\(-0.0236242\pi\)
\(47\)	−12250.2	−	12250.2i	−0.808904	−	0.808904i	0.175564	−	0.984468i	\(-0.443825\pi\)
							−0.984468	+	0.175564i	\(0.943825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.6	yes	116
5.2	odd	4		80.6.j.a.67.24	yes	116
16.11	odd	4		80.6.j.a.43.24	✓	116
80.27	even	4	inner	80.6.s.a.27.6	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.24	✓	116	16.11	odd	4
80.6.j.a.67.24	yes	116	5.2	odd	4
80.6.s.a.3.6	yes	116	1.1	even	1	trivial
80.6.s.a.27.6	yes	116	80.27	even	4	inner