Embedded newform 80.6.j.a.43.8

• Label: 80.6.j.a.43.8
• 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			43.8
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.39225 - 1.70986i) q^{2} -25.2540i q^{3} +(26.1528 + 18.4400i) q^{4} +(44.1288 - 34.3168i) q^{5} +(-43.1808 + 136.176i) q^{6} +(-154.305 + 154.305i) q^{7} +(-109.492 - 144.151i) q^{8} -394.763 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.39225	−	1.70986i	−0.953224	−	0.302263i
							\(3\)		−	25.2540i		−	1.62004i	−0.586400	−	0.810022i	\(-0.699455\pi\)
0.586400	−	0.810022i	\(-0.300545\pi\)
\(5\)	44.1288	−	34.3168i	0.789401	−	0.613878i
							\(7\)	−154.305	+	154.305i	−1.19024	+	1.19024i	−0.213237	+	0.977000i	\(0.568401\pi\)
−0.977000	+	0.213237i	\(0.931599\pi\)
\(11\)	200.618	−	200.618i	0.499906	−	0.499906i	−0.411503	−	0.911409i	\(-0.634996\pi\)
							0.911409	+	0.411503i	\(0.134996\pi\)
\(13\)	−579.080			−0.950343			−0.475172	−	0.879893i	\(-0.657614\pi\)
							−0.475172	+	0.879893i	\(0.657614\pi\)
\(17\)	−542.766	+	542.766i	−0.455502	+	0.455502i	−0.897176	−	0.441674i	\(-0.854385\pi\)
							0.441674	+	0.897176i	\(0.354385\pi\)
\(19\)	−1636.29	+	1636.29i	−1.03987	+	1.03987i	−0.0406944	+	0.999172i	\(0.512957\pi\)
							−0.999172	+	0.0406944i	\(0.987043\pi\)
\(23\)	−1455.35	−	1455.35i	−0.573652	−	0.573652i	0.359495	−	0.933147i	\(-0.382949\pi\)
							−0.933147	+	0.359495i	\(0.882949\pi\)
\(29\)	2832.77	+	2832.77i	0.625485	+	0.625485i	0.946929	−	0.321444i	\(-0.104168\pi\)
							−0.321444	+	0.946929i	\(0.604168\pi\)
\(31\)			4683.34i			0.875289i	0.899148	+	0.437645i	\(0.144187\pi\)
							−0.899148	+	0.437645i	\(0.855813\pi\)
\(37\)	12400.1			1.48909			0.744547	−	0.667570i	\(-0.232666\pi\)
							0.744547	+	0.667570i	\(0.232666\pi\)
\(41\)			1881.91i			0.174840i	0.996172	+	0.0874199i	\(0.0278622\pi\)
							−0.996172	+	0.0874199i	\(0.972138\pi\)
\(43\)	−12845.0			−1.05941			−0.529703	−	0.848183i	\(-0.677697\pi\)
							−0.529703	+	0.848183i	\(0.677697\pi\)
\(47\)	−6084.99	−	6084.99i	−0.401805	−	0.401805i	0.477064	−	0.878869i	\(-0.341701\pi\)
							−0.878869	+	0.477064i	\(0.841701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.8	✓	116
5.2	odd	4		80.6.s.a.27.36	yes	116
16.3	odd	4		80.6.s.a.3.36	yes	116
80.67	even	4	inner	80.6.j.a.67.8	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.8	✓	116	1.1	even	1	trivial
80.6.j.a.67.8	yes	116	80.67	even	4	inner
80.6.s.a.3.36	yes	116	16.3	odd	4
80.6.s.a.27.36	yes	116	5.2	odd	4