Embedded newform 605.2.s.a.16.17

• Label: 605.2.s.a.16.17
• 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			16.17
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.66802 + 0.191388i) q^{2} +(-1.24319 - 0.903228i) q^{3} +(0.797652 + 0.185486i) q^{4} +(0.564443 - 0.825472i) q^{5} +(-1.90080 - 1.74454i) q^{6} +(3.32658 + 3.42297i) q^{7} +(-1.86765 - 0.666376i) q^{8} +(-0.197358 - 0.607406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.66802	+	0.191388i	1.17947	+	0.135332i	0.681471	−	0.731845i	\(-0.261341\pi\)
							0.498000	+	0.867177i	\(0.334068\pi\)
\(3\)	−1.24319	−	0.903228i	−0.717754	−	0.521479i	0.167912	−	0.985802i	\(-0.446298\pi\)
							−0.885666	+	0.464323i	\(0.846298\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	3.32658	+	3.42297i	1.25733	+	1.29376i	0.938219	+	0.346043i	\(0.112475\pi\)
0.319110	+	0.947718i	\(0.396616\pi\)
\(11\)	3.12642	+	1.10703i	0.942651	+	0.333781i
							\(13\)	2.31829	−	3.84445i	0.642978	−	1.06626i	−0.348665	−	0.937247i	\(-0.613365\pi\)
0.991643	−	0.129011i	\(-0.0411804\pi\)
\(17\)	2.10953	−	1.72495i	0.511635	−	0.418362i	−0.341634	−	0.939833i	\(-0.610980\pi\)
							0.853269	+	0.521471i	\(0.174617\pi\)
\(19\)	1.10971	+	0.0634555i	0.254585	+	0.0145577i	0.183915	−	0.982942i	\(-0.441123\pi\)
							0.0706696	+	0.997500i	\(0.477486\pi\)
\(23\)	0.147273	−	1.02431i	0.0307086	−	0.213583i	−0.968689	−	0.248276i	\(-0.920136\pi\)
							0.999398	+	0.0346927i	\(0.0110453\pi\)
\(29\)	1.54655	−	3.97229i	0.287188	−	0.737636i	−0.712177	−	0.702000i	\(-0.752291\pi\)
							0.999365	−	0.0356358i	\(-0.0113456\pi\)
\(31\)	1.37245	+	0.580002i	0.246499	+	0.104172i	0.509019	−	0.860755i	\(-0.330008\pi\)
							−0.262520	+	0.964927i	\(0.584554\pi\)
\(37\)	0.302771	−	3.52509i	0.0497752	−	0.579522i	−0.928378	−	0.371637i	\(-0.878797\pi\)
							0.978153	−	0.207885i	\(-0.0666579\pi\)
\(41\)	−1.42259	+	0.803375i	−0.222172	+	0.125466i	−0.598801	−	0.800898i	\(-0.704356\pi\)
							0.376629	+	0.926364i	\(0.377083\pi\)
\(43\)	5.91753	−	1.73754i	0.902415	−	0.264973i	0.202570	−	0.979268i	\(-0.435071\pi\)
							0.699845	+	0.714295i	\(0.253252\pi\)
\(47\)	−1.86551	+	9.20665i	−0.272113	+	1.34293i	0.577875	+	0.816125i	\(0.303882\pi\)
							−0.849987	+	0.526803i	\(0.823391\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.17	✓	880
121.53	even	55	inner	605.2.s.a.416.17	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.17	✓	880	1.1	even	1	trivial
605.2.s.a.416.17	yes	880	121.53	even	55	inner