Embedded newform 80.2.j.b.43.5

• Label: 80.2.j.b.43.5
• Level: $80$
• Weight: $2$
• Character: 80.43
• Analytic conductor: $0.639$
• Analytic rank: $0$
• Dimension: $18$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	80.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.638803216170\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(9\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 2*x^16 - 4*x^15 - 5*x^14 - 14*x^13 - 10*x^12 + 6*x^11 + 37*x^10 + 70*x^9 + 74*x^8 + 24*x^7 - 80*x^6 - 224*x^5 - 160*x^4 - 256*x^3 + 256*x^2 + 512
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.5
Root			\(-0.480367 - 1.33013i\) of defining polynomial
Character	\(\chi\)	\(=\)	80.43
Dual form			80.2.j.b.67.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.307817 + 1.38031i) q^{2} +2.85601i q^{3} +(-1.81050 - 0.849763i) q^{4} +(1.43498 - 1.71489i) q^{5} +(-3.94217 - 0.879127i) q^{6} +(-0.458895 + 0.458895i) q^{7} +(1.73024 - 2.23747i) q^{8} -5.15678 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 8 q^{6} + 2 q^{7} - 4 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.307817	+	1.38031i	−0.217659	+	0.976025i
							\(3\)			2.85601i			1.64892i	0.565923	+	0.824458i	\(0.308520\pi\)
−0.565923	+	0.824458i	\(0.691480\pi\)
\(5\)	1.43498	−	1.71489i	0.641741	−	0.766921i
							\(7\)	−0.458895	+	0.458895i	−0.173446	+	0.173446i	−0.788491	−	0.615046i	\(-0.789138\pi\)
0.615046	+	0.788491i	\(0.289138\pi\)
\(11\)	−0.492763	+	0.492763i	−0.148574	+	0.148574i	−0.777481	−	0.628907i	\(-0.783503\pi\)
							0.628907	+	0.777481i	\(0.283503\pi\)
\(13\)	4.52109			1.25393			0.626963	−	0.779049i	\(-0.284298\pi\)
							0.626963	+	0.779049i	\(0.284298\pi\)
\(17\)	−3.12823	+	3.12823i	−0.758708	+	0.758708i	−0.976087	−	0.217379i	\(-0.930249\pi\)
							0.217379	+	0.976087i	\(0.430249\pi\)
\(19\)	4.04508	−	4.04508i	0.928005	−	0.928005i	−0.0695721	−	0.997577i	\(-0.522163\pi\)
							0.997577	+	0.0695721i	\(0.0221634\pi\)
\(23\)	−1.80660	−	1.80660i	−0.376701	−	0.376701i	0.493209	−	0.869911i	\(-0.335824\pi\)
							−0.869911	+	0.493209i	\(0.835824\pi\)
\(29\)	−3.83926	−	3.83926i	−0.712932	−	0.712932i	0.254215	−	0.967148i	\(-0.418183\pi\)
							−0.967148	+	0.254215i	\(0.918183\pi\)
\(31\)		−	0.139949i		−	0.0251356i	−0.999921	−	0.0125678i	\(-0.995999\pi\)
							0.999921	−	0.0125678i	\(-0.00400057\pi\)
\(37\)	5.84330			0.960633			0.480317	−	0.877095i	\(-0.340522\pi\)
							0.480317	+	0.877095i	\(0.340522\pi\)
\(41\)		−	4.55648i		−	0.711602i	−0.934562	−	0.355801i	\(-0.884208\pi\)
							0.934562	−	0.355801i	\(-0.115792\pi\)
\(43\)	−7.49928			−1.14363			−0.571815	−	0.820383i	\(-0.693760\pi\)
							−0.571815	+	0.820383i	\(0.693760\pi\)
\(47\)	−4.14073	−	4.14073i	−0.603987	−	0.603987i	0.337381	−	0.941368i	\(-0.390459\pi\)
							−0.941368	+	0.337381i	\(0.890459\pi\)