Embedded newform 80.2.s.b.3.5

• Label: 80.2.s.b.3.5
• Level: $80$
• Weight: $2$
• Character: 80.3
• 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.s (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			3.5
Root			\(0.235136 - 1.39453i\) of defining polynomial
Character	\(\chi\)	\(=\)	80.3
Dual form			80.2.s.b.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.430311 + 1.34716i) q^{2} -2.96561 q^{3} +(-1.62967 + 1.15939i) q^{4} +(-0.177336 + 2.22902i) q^{5} +(-1.27613 - 3.99515i) q^{6} +(-0.115101 + 0.115101i) q^{7} +(-2.26315 - 1.69652i) q^{8} +5.79486 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.430311	+	1.34716i	0.304276	+	0.952584i
							\(3\)	−2.96561			−1.71220			−0.856099	−	0.516813i	\(-0.827118\pi\)
−0.856099	+	0.516813i	\(0.827118\pi\)
\(5\)	−0.177336	+	2.22902i	−0.0793073	+	0.996850i
							\(7\)	−0.115101	+	0.115101i	−0.0435040	+	0.0435040i	−0.728524	−	0.685020i	\(-0.759793\pi\)
0.685020	+	0.728524i	\(0.259793\pi\)
\(11\)	2.95966	+	2.95966i	0.892372	+	0.892372i	0.994746	−	0.102374i	\(-0.0326439\pi\)
							−0.102374	+	0.994746i	\(0.532644\pi\)
\(13\)			1.55822i			0.432172i	0.976374	+	0.216086i	\(0.0693292\pi\)
							−0.976374	+	0.216086i	\(0.930671\pi\)
\(17\)	0.299668	−	0.299668i	0.0726801	−	0.0726801i	−0.669832	−	0.742512i	\(-0.733634\pi\)
							0.742512	+	0.669832i	\(0.233634\pi\)
\(19\)	−2.26261	−	2.26261i	−0.519079	−	0.519079i	0.398214	−	0.917293i	\(-0.369630\pi\)
							−0.917293	+	0.398214i	\(0.869630\pi\)
\(23\)	4.14573	+	4.14573i	0.864444	+	0.864444i	0.991851	−	0.127406i	\(-0.0406652\pi\)
							−0.127406	+	0.991851i	\(0.540665\pi\)
\(29\)	0.289656	−	0.289656i	0.0537878	−	0.0537878i	−0.679701	−	0.733489i	\(-0.737891\pi\)
							0.733489	+	0.679701i	\(0.237891\pi\)
\(31\)			4.18508i			0.751663i	0.926688	+	0.375832i	\(0.122643\pi\)
							−0.926688	+	0.375832i	\(0.877357\pi\)
\(37\)		−	1.63643i		−	0.269027i	−0.990912	−	0.134514i	\(-0.957053\pi\)
							0.990912	−	0.134514i	\(-0.0429472\pi\)
\(41\)		−	7.61648i		−	1.18949i	−0.803913	−	0.594747i	\(-0.797252\pi\)
							0.803913	−	0.594747i	\(-0.202748\pi\)
\(43\)		−	6.72651i		−	1.02578i	−0.858453	−	0.512892i	\(-0.828574\pi\)
							0.858453	−	0.512892i	\(-0.171426\pi\)
\(47\)	4.38366	+	4.38366i	0.639423	+	0.639423i	0.950413	−	0.310990i	\(-0.100661\pi\)
							−0.310990	+	0.950413i	\(0.600661\pi\)