Embedded newform 605.2.s.a.16.18

• Label: 605.2.s.a.16.18
• 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			16.18
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.90115 + 0.218137i) q^{2} +(1.85412 + 1.34710i) q^{3} +(1.61876 + 0.376427i) q^{4} +(0.564443 - 0.825472i) q^{5} +(3.23112 + 2.96549i) q^{6} +(0.529131 + 0.544463i) q^{7} +(-0.609275 - 0.217389i) q^{8} +(0.696047 + 2.14221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.90115	+	0.218137i	1.34432	+	0.154246i	0.756099	−	0.654457i	\(-0.227103\pi\)
							0.588217	+	0.808703i	\(0.299830\pi\)
\(3\)	1.85412	+	1.34710i	1.07048	+	0.777749i	0.975998	−	0.217778i	\(-0.0698809\pi\)
							0.0944807	+	0.995527i	\(0.469881\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	0.529131	+	0.544463i	0.199993	+	0.205788i	0.809708	−	0.586833i	\(-0.199625\pi\)
−0.609715	+	0.792620i	\(0.708716\pi\)
\(11\)	3.19945	−	0.873778i	0.964672	−	0.263454i
							\(13\)	−0.206773	+	0.342895i	−0.0573487	+	0.0951021i	−0.883702	−	0.468050i	\(-0.844957\pi\)
0.826353	+	0.563152i	\(0.190411\pi\)
\(17\)	−2.47560	+	2.02429i	−0.600421	+	0.490962i	−0.883686	−	0.468080i	\(-0.844946\pi\)
							0.283265	+	0.959042i	\(0.408582\pi\)
\(19\)	−4.23254	−	0.242025i	−0.971011	−	0.0555244i	−0.435622	−	0.900130i	\(-0.643471\pi\)
							−0.535389	+	0.844605i	\(0.679835\pi\)
\(23\)	0.0440747	−	0.306546i	0.00919021	−	0.0639193i	−0.984710	−	0.174200i	\(-0.944266\pi\)
							0.993900	+	0.110281i	\(0.0351751\pi\)
\(29\)	−2.25871	+	5.80145i	−0.419432	+	1.07730i	0.550696	+	0.834706i	\(0.314362\pi\)
							−0.970128	+	0.242595i	\(0.922001\pi\)
\(31\)	0.835239	+	0.352975i	0.150013	+	0.0633962i	0.462984	−	0.886367i	\(-0.346779\pi\)
							−0.312971	+	0.949763i	\(0.601324\pi\)
\(37\)	0.161433	−	1.87953i	0.0265395	−	0.308993i	−0.971011	−	0.239037i	\(-0.923168\pi\)
							0.997550	−	0.0699568i	\(-0.0222862\pi\)
\(41\)	0.0950575	−	0.0536814i	0.0148455	−	0.00838363i	−0.484292	−	0.874906i	\(-0.660923\pi\)
							0.499138	+	0.866523i	\(0.333650\pi\)
\(43\)	−1.48522	+	0.436100i	−0.226494	+	0.0665046i	−0.393010	−	0.919534i	\(-0.628566\pi\)
							0.166516	+	0.986039i	\(0.446748\pi\)
\(47\)	0.818038	−	4.03718i	0.119323	−	0.588883i	−0.875147	−	0.483858i	\(-0.839235\pi\)
							0.994470	−	0.105025i	\(-0.0334921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.18	✓	880
121.53	even	55	inner	605.2.s.a.416.18	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.18	✓	880	1.1	even	1	trivial
605.2.s.a.416.18	yes	880	121.53	even	55	inner