Embedded newform 80.6.q.a.69.11

• Label: 80.6.q.a.69.11
• Level: $80$
• Weight: $6$
• Character: 80.69
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.q (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			69.11
Character	\(\chi\)	\(=\)	80.69
Dual form			80.6.q.a.29.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.74064 + 3.08648i) q^{2} +(-1.81560 - 1.81560i) q^{3} +(12.9473 - 29.2637i) q^{4} +(48.1357 + 28.4245i) q^{5} +(14.2109 + 3.00330i) q^{6} -66.7879 q^{7} +(28.9433 + 178.690i) q^{8} -236.407i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 4 q^{4} - 2 q^{5} - 4 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.74064	+	3.08648i	−0.838035	+	0.545617i
							\(3\)	−1.81560	−	1.81560i	−0.116471	−	0.116471i	0.646469	−	0.762940i	\(-0.276245\pi\)
−0.762940	+	0.646469i	\(0.776245\pi\)
\(5\)	48.1357	+	28.4245i	0.861078	+	0.508472i
							\(7\)	−66.7879			−0.515173			−0.257586	−	0.966255i	\(-0.582927\pi\)
−0.257586	+	0.966255i	\(0.582927\pi\)
\(11\)	215.262	+	215.262i	0.536397	+	0.536397i	0.922469	−	0.386072i	\(-0.126168\pi\)
							−0.386072	+	0.922469i	\(0.626168\pi\)
\(13\)	−631.811	−	631.811i	−1.03688	−	1.03688i	−0.999293	−	0.0375869i	\(-0.988033\pi\)
							−0.0375869	−	0.999293i	\(-0.511967\pi\)
\(17\)			755.437i			0.633981i	0.948429	+	0.316990i	\(0.102672\pi\)
							−0.948429	+	0.316990i	\(0.897328\pi\)
\(19\)	1909.08	−	1909.08i	1.21322	−	1.21322i	0.243262	−	0.969960i	\(-0.421782\pi\)
							0.969960	−	0.243262i	\(-0.0782176\pi\)
\(23\)	3963.77			1.56239			0.781194	−	0.624288i	\(-0.214611\pi\)
							0.781194	+	0.624288i	\(0.214611\pi\)
\(29\)	586.717	−	586.717i	0.129549	−	0.129549i	−0.639359	−	0.768908i	\(-0.720800\pi\)
							0.768908	+	0.639359i	\(0.220800\pi\)
\(31\)	4694.91			0.877451			0.438725	−	0.898621i	\(-0.355430\pi\)
							0.438725	+	0.898621i	\(0.355430\pi\)
\(37\)	9100.34	−	9100.34i	1.09283	−	1.09283i	0.0976066	−	0.995225i	\(-0.468881\pi\)
							0.995225	−	0.0976066i	\(-0.0311187\pi\)
\(41\)		−	4883.98i		−	0.453747i	−0.973924	−	0.226874i	\(-0.927150\pi\)
							0.973924	−	0.226874i	\(-0.0728504\pi\)
\(43\)	6120.84	−	6120.84i	0.504824	−	0.504824i	−0.408109	−	0.912933i	\(-0.633812\pi\)
							0.912933	+	0.408109i	\(0.133812\pi\)
\(47\)		−	17818.7i		−	1.17661i	−0.808641	−	0.588303i	\(-0.799796\pi\)
							0.808641	−	0.588303i	\(-0.200204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.q.a.69.11	yes	116
5.4	even	2	inner	80.6.q.a.69.48	yes	116
16.13	even	4	inner	80.6.q.a.29.48	yes	116
80.29	even	4	inner	80.6.q.a.29.11	✓	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.q.a.29.11	✓	116	80.29	even	4	inner
80.6.q.a.29.48	yes	116	16.13	even	4	inner
80.6.q.a.69.11	yes	116	1.1	even	1	trivial
80.6.q.a.69.48	yes	116	5.4	even	2	inner