Embedded newform 605.2.s.b.16.4

• Label: 605.2.s.b.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80070 - 0.206611i) q^{2} +(-1.29126 - 0.938155i) q^{3} +(1.25182 + 0.291098i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(2.13134 + 1.95613i) q^{6} +(-0.0685099 - 0.0704951i) q^{7} +(1.22021 + 0.435369i) q^{8} +(-0.139834 - 0.430365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80070	−	0.206611i	−1.27329	−	0.146096i	−0.549135	−	0.835734i	\(-0.685043\pi\)
							−0.724155	+	0.689637i	\(0.757770\pi\)
\(3\)	−1.29126	−	0.938155i	−0.745509	−	0.541644i	0.148922	−	0.988849i	\(-0.452420\pi\)
							−0.894432	+	0.447205i	\(0.852420\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−0.0685099	−	0.0704951i	−0.0258943	−	0.0266446i	0.704061	−	0.710140i	\(-0.251368\pi\)
−0.729955	+	0.683495i	\(0.760459\pi\)
\(11\)	−0.376086	−	3.29523i	−0.113394	−	0.993550i
							\(13\)	−2.20746	+	3.66065i	−0.612238	+	1.01528i	0.383543	+	0.923523i	\(0.374704\pi\)
−0.995781	+	0.0917599i	\(0.970751\pi\)
\(17\)	−3.22476	+	2.63687i	−0.782118	+	0.639535i	−0.937368	−	0.348340i	\(-0.886745\pi\)
							0.155250	+	0.987875i	\(0.450382\pi\)
\(19\)	−0.263287	−	0.0150553i	−0.0604021	−	0.00345392i	0.0268616	−	0.999639i	\(-0.491449\pi\)
							−0.0872637	+	0.996185i	\(0.527812\pi\)
\(23\)	0.379754	−	2.64125i	0.0791842	−	0.550738i	−0.911154	−	0.412066i	\(-0.864807\pi\)
							0.990338	−	0.138673i	\(-0.0442835\pi\)
\(29\)	−0.892183	+	2.29155i	−0.165674	+	0.425531i	−0.990087	−	0.140459i	\(-0.955142\pi\)
							0.824412	+	0.565990i	\(0.191506\pi\)
\(31\)	5.76378	+	2.43580i	1.03521	+	0.437482i	0.839603	−	0.543201i	\(-0.182788\pi\)
							0.195604	+	0.980683i	\(0.437333\pi\)
\(37\)	−0.824959	+	9.60482i	−0.135622	+	1.57902i	0.539930	+	0.841710i	\(0.318451\pi\)
							−0.675552	+	0.737312i	\(0.736095\pi\)
\(41\)	5.23917	−	2.95869i	0.818220	−	0.462070i	−0.0249682	−	0.999688i	\(-0.507948\pi\)
							0.843189	+	0.537618i	\(0.180676\pi\)
\(43\)	4.86190	−	1.42758i	0.741433	−	0.217704i	0.110862	−	0.993836i	\(-0.464639\pi\)
							0.630571	+	0.776132i	\(0.282821\pi\)
\(47\)	2.13115	−	10.5176i	0.310860	−	1.53415i	−0.454616	−	0.890688i	\(-0.650223\pi\)
							0.765475	−	0.643465i	\(-0.222504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.4	✓	880
121.53	even	55	inner	605.2.s.b.416.4	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.4	✓	880	1.1	even	1	trivial
605.2.s.b.416.4	yes	880	121.53	even	55	inner