Embedded newform 80.6.s.a.3.3

• Label: 80.6.s.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.58900 + 0.873525i) q^{2} -13.7617 q^{3} +(30.4739 - 9.76427i) q^{4} +(-55.0423 - 9.76467i) q^{5} +(76.9143 - 12.0212i) q^{6} +(-58.2575 + 58.2575i) q^{7} +(-161.789 + 81.1923i) q^{8} -53.6151 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.58900	+	0.873525i	−0.988005	+	0.154419i
							\(3\)	−13.7617			−0.882815			−0.441407	−	0.897307i	\(-0.645521\pi\)
−0.441407	+	0.897307i	\(0.645521\pi\)
\(5\)	−55.0423	−	9.76467i	−0.984626	−	0.174676i
							\(7\)	−58.2575	+	58.2575i	−0.449373	+	0.449373i	−0.895146	−	0.445773i	\(-0.852929\pi\)
0.445773	+	0.895146i	\(0.352929\pi\)
\(11\)	−480.469	−	480.469i	−1.19725	−	1.19725i	−0.974987	−	0.222260i	\(-0.928657\pi\)
							−0.222260	−	0.974987i	\(-0.571343\pi\)
\(13\)		−	129.330i		−	0.212246i	−0.994353	−	0.106123i	\(-0.966156\pi\)
							0.994353	−	0.106123i	\(-0.0338438\pi\)
\(17\)	−889.454	+	889.454i	−0.746451	+	0.746451i	−0.973811	−	0.227360i	\(-0.926991\pi\)
							0.227360	+	0.973811i	\(0.426991\pi\)
\(19\)	570.032	+	570.032i	0.362256	+	0.362256i	0.864643	−	0.502387i	\(-0.167545\pi\)
							−0.502387	+	0.864643i	\(0.667545\pi\)
\(23\)	1641.50	+	1641.50i	0.647024	+	0.647024i	0.952273	−	0.305248i	\(-0.0987395\pi\)
							−0.305248	+	0.952273i	\(0.598739\pi\)
\(29\)	3201.82	−	3201.82i	0.706972	−	0.706972i	−0.258925	−	0.965897i	\(-0.583368\pi\)
							0.965897	+	0.258925i	\(0.0833682\pi\)
\(31\)		−	4355.95i		−	0.814102i	−0.913406	−	0.407051i	\(-0.866557\pi\)
							0.913406	−	0.407051i	\(-0.133443\pi\)
\(37\)			10428.0i			1.25226i	0.779717	+	0.626132i	\(0.215363\pi\)
							−0.779717	+	0.626132i	\(0.784637\pi\)
\(41\)			4323.69i			0.401694i	0.979623	+	0.200847i	\(0.0643694\pi\)
							−0.979623	+	0.200847i	\(0.935631\pi\)
\(43\)		−	8102.97i		−	0.668303i	−0.942519	−	0.334151i	\(-0.891550\pi\)
							0.942519	−	0.334151i	\(-0.108450\pi\)
\(47\)	−16761.5	−	16761.5i	−1.10680	−	1.10680i	−0.993569	−	0.113230i	\(-0.963880\pi\)
							−0.113230	−	0.993569i	\(-0.536120\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.3	yes	116
5.2	odd	4		80.6.j.a.67.25	yes	116
16.11	odd	4		80.6.j.a.43.25	✓	116
80.27	even	4	inner	80.6.s.a.27.3	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.25	✓	116	16.11	odd	4
80.6.j.a.67.25	yes	116	5.2	odd	4
80.6.s.a.3.3	yes	116	1.1	even	1	trivial
80.6.s.a.27.3	yes	116	80.27	even	4	inner