Embedded newform 80.6.j.a.43.6

• Label: 80.6.j.a.43.6
• Level: $80$
• Weight: $6$
• Character: 80.43
• 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.j (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			43.6
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.48456 + 1.38551i) q^{2} +7.86510i q^{3} +(28.1607 - 15.1978i) q^{4} +(34.2061 + 44.2148i) q^{5} +(-10.8972 - 43.1366i) q^{6} +(-94.0660 + 94.0660i) q^{7} +(-133.392 + 122.370i) q^{8} +181.140 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.48456	+	1.38551i	−0.969542	+	0.244926i
							\(3\)			7.86510i			0.504546i	0.967656	+	0.252273i	\(0.0811782\pi\)
−0.967656	+	0.252273i	\(0.918822\pi\)
\(5\)	34.2061	+	44.2148i	0.611897	+	0.790938i
							\(7\)	−94.0660	+	94.0660i	−0.725584	+	0.725584i	−0.969737	−	0.244153i	\(-0.921490\pi\)
0.244153	+	0.969737i	\(0.421490\pi\)
\(11\)	263.645	−	263.645i	0.656959	−	0.656959i	−0.297701	−	0.954659i	\(-0.596220\pi\)
							0.954659	+	0.297701i	\(0.0962198\pi\)
\(13\)	1002.78			1.64569			0.822843	−	0.568269i	\(-0.192387\pi\)
							0.822843	+	0.568269i	\(0.192387\pi\)
\(17\)	−1214.31	+	1214.31i	−1.01908	+	1.01908i	−0.0192636	+	0.999814i	\(0.506132\pi\)
							−0.999814	+	0.0192636i	\(0.993868\pi\)
\(19\)	−1148.12	+	1148.12i	−0.729631	+	0.729631i	−0.970546	−	0.240915i	\(-0.922553\pi\)
							0.240915	+	0.970546i	\(0.422553\pi\)
\(23\)	−1457.80	−	1457.80i	−0.574618	−	0.574618i	0.358798	−	0.933415i	\(-0.383187\pi\)
							−0.933415	+	0.358798i	\(0.883187\pi\)
\(29\)	184.895	+	184.895i	0.0408254	+	0.0408254i	0.727225	−	0.686399i	\(-0.240810\pi\)
							−0.686399	+	0.727225i	\(0.740810\pi\)
\(31\)			1912.98i			0.357524i	0.983892	+	0.178762i	\(0.0572092\pi\)
							−0.983892	+	0.178762i	\(0.942791\pi\)
\(37\)	−9982.06			−1.19871			−0.599357	−	0.800482i	\(-0.704577\pi\)
							−0.599357	+	0.800482i	\(0.704577\pi\)
\(41\)			8423.52i			0.782590i	0.920265	+	0.391295i	\(0.127973\pi\)
							−0.920265	+	0.391295i	\(0.872027\pi\)
\(43\)	6371.50			0.525497			0.262749	−	0.964864i	\(-0.415371\pi\)
							0.262749	+	0.964864i	\(0.415371\pi\)
\(47\)	−1779.81	−	1779.81i	−0.117525	−	0.117525i	0.645898	−	0.763423i	\(-0.276483\pi\)
							−0.763423	+	0.645898i	\(0.776483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.6	✓	116
5.2	odd	4		80.6.s.a.27.24	yes	116
16.3	odd	4		80.6.s.a.3.24	yes	116
80.67	even	4	inner	80.6.j.a.67.6	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.6	✓	116	1.1	even	1	trivial
80.6.j.a.67.6	yes	116	80.67	even	4	inner
80.6.s.a.3.24	yes	116	16.3	odd	4
80.6.s.a.27.24	yes	116	5.2	odd	4