Embedded newform 605.2.s.a.16.20

• Label: 605.2.s.a.16.20
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.20
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46804 + 0.283181i) q^{2} +(2.06854 + 1.50288i) q^{3} +(4.06299 + 0.944807i) q^{4} +(0.564443 - 0.825472i) q^{5} +(4.67964 + 4.29494i) q^{6} +(-3.42528 - 3.52452i) q^{7} +(5.08053 + 1.81273i) q^{8} +(1.09315 + 3.36437i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46804	+	0.283181i	1.74516	+	0.200239i	0.927081	−	0.374860i	\(-0.122309\pi\)
							0.818084	+	0.575099i	\(0.195037\pi\)
\(3\)	2.06854	+	1.50288i	1.19427	+	0.867689i	0.993709	−	0.111992i	\(-0.0357231\pi\)
							0.200562	+	0.979681i	\(0.435723\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	−3.42528	−	3.52452i	−1.29463	−	1.33215i	−0.913070	−	0.407802i	\(-0.866295\pi\)
−0.381562	−	0.924343i	\(-0.624614\pi\)
\(11\)	−3.20187	+	0.864877i	−0.965401	+	0.260770i
							\(13\)	−2.89131	+	4.79470i	−0.801904	+	1.32981i	0.138710	+	0.990333i	\(0.455704\pi\)
−0.940615	+	0.339476i	\(0.889750\pi\)
\(17\)	3.95442	−	3.23351i	0.959087	−	0.784242i	−0.0173929	−	0.999849i	\(-0.505537\pi\)
							0.976480	+	0.215607i	\(0.0691730\pi\)
\(19\)	−0.149808	−	0.00856636i	−0.0343684	−	0.00196526i	0.0398961	−	0.999204i	\(-0.487297\pi\)
							−0.0742646	+	0.997239i	\(0.523661\pi\)
\(23\)	−0.768065	+	5.34201i	−0.160153	+	1.11389i	0.738191	+	0.674592i	\(0.235680\pi\)
							−0.898344	+	0.439294i	\(0.855229\pi\)
\(29\)	1.68252	−	4.32153i	0.312437	−	0.802488i	−0.684837	−	0.728697i	\(-0.740126\pi\)
							0.997274	−	0.0737914i	\(-0.0235099\pi\)
\(31\)	1.89094	+	0.799116i	0.339622	+	0.143525i	0.552409	−	0.833573i	\(-0.313709\pi\)
							−0.212787	+	0.977099i	\(0.568254\pi\)
\(37\)	0.623818	−	7.26298i	0.102555	−	1.19403i	−0.746377	−	0.665524i	\(-0.768208\pi\)
							0.848932	−	0.528502i	\(-0.177246\pi\)
\(41\)	−4.56627	+	2.57869i	−0.713131	+	0.402724i	−0.805180	−	0.593031i	\(-0.797931\pi\)
							0.0920485	+	0.995755i	\(0.470659\pi\)
\(43\)	11.6075	−	3.40828i	1.77013	−	0.519758i	0.776272	−	0.630399i	\(-0.217109\pi\)
							0.993860	+	0.110641i	\(0.0352904\pi\)
\(47\)	−0.995059	+	4.91081i	−0.145144	+	0.716315i	0.839724	+	0.543013i	\(0.182717\pi\)
							−0.984869	+	0.173303i	\(0.944556\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.20	✓	880
121.53	even	55	inner	605.2.s.a.416.20	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.20	✓	880	1.1	even	1	trivial
605.2.s.a.416.20	yes	880	121.53	even	55	inner