Embedded newform 80.6.s.a.3.4

• Label: 80.6.s.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.54990 - 1.09479i) q^{2} -11.0755 q^{3} +(29.6029 + 12.1520i) q^{4} +(24.0182 + 50.4790i) q^{5} +(61.4682 + 12.1254i) q^{6} +(119.074 - 119.074i) q^{7} +(-150.989 - 99.8515i) q^{8} -120.332 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.54990	−	1.09479i	−0.981094	−	0.193534i
							\(3\)	−11.0755			−0.710496			−0.355248	−	0.934772i	\(-0.615604\pi\)
−0.355248	+	0.934772i	\(0.615604\pi\)
\(5\)	24.0182	+	50.4790i	0.429650	+	0.902996i
							\(7\)	119.074	−	119.074i	0.918484	−	0.918484i	−0.0784351	−	0.996919i	\(-0.524992\pi\)
0.996919	+	0.0784351i	\(0.0249923\pi\)
\(11\)	−284.172	−	284.172i	−0.708109	−	0.708109i	0.258029	−	0.966137i	\(-0.416927\pi\)
							−0.966137	+	0.258029i	\(0.916927\pi\)
\(13\)			895.419i			1.46950i	0.678341	+	0.734748i	\(0.262699\pi\)
							−0.678341	+	0.734748i	\(0.737301\pi\)
\(17\)	1464.97	−	1464.97i	1.22944	−	1.22944i	0.265264	−	0.964176i	\(-0.414541\pi\)
							0.964176	−	0.265264i	\(-0.0854590\pi\)
\(19\)	504.802	+	504.802i	0.320802	+	0.320802i	0.849075	−	0.528273i	\(-0.177160\pi\)
							−0.528273	+	0.849075i	\(0.677160\pi\)
\(23\)	1554.00	+	1554.00i	0.612534	+	0.612534i	0.943606	−	0.331071i	\(-0.107410\pi\)
							−0.331071	+	0.943606i	\(0.607410\pi\)
\(29\)	−4052.56	+	4052.56i	−0.894817	+	0.894817i	−0.994972	−	0.100155i	\(-0.968066\pi\)
							0.100155	+	0.994972i	\(0.468066\pi\)
\(31\)			6835.00i			1.27742i	0.769447	+	0.638711i	\(0.220532\pi\)
							−0.769447	+	0.638711i	\(0.779468\pi\)
\(37\)			8790.81i			1.05566i	0.849350	+	0.527830i	\(0.176994\pi\)
							−0.849350	+	0.527830i	\(0.823006\pi\)
\(41\)		−	5567.50i		−	0.517250i	−0.965978	−	0.258625i	\(-0.916731\pi\)
							0.965978	−	0.258625i	\(-0.0832694\pi\)
\(43\)			8918.06i			0.735528i	0.929919	+	0.367764i	\(0.119877\pi\)
							−0.929919	+	0.367764i	\(0.880123\pi\)
\(47\)	17092.7	+	17092.7i	1.12867	+	1.12867i	0.990394	+	0.138276i	\(0.0441562\pi\)
							0.138276	+	0.990394i	\(0.455844\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.4	yes	116
5.2	odd	4		80.6.j.a.67.32	yes	116
16.11	odd	4		80.6.j.a.43.32	✓	116
80.27	even	4	inner	80.6.s.a.27.4	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.32	✓	116	16.11	odd	4
80.6.j.a.67.32	yes	116	5.2	odd	4
80.6.s.a.3.4	yes	116	1.1	even	1	trivial
80.6.s.a.27.4	yes	116	80.27	even	4	inner