Embedded newform 605.2.k.b.56.16

• Label: 605.2.k.b.56.16
• Level: $605$
• Weight: $2$
• Character: 605.56
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			56.16
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.59296 + 0.467736i) q^{2} -2.26440 q^{3} +(0.636241 + 0.408888i) q^{4} +(0.654861 - 0.755750i) q^{5} +(-3.60711 - 1.05914i) q^{6} +(-0.226450 + 0.495857i) q^{7} +(-1.35216 - 1.56047i) q^{8} +2.12752 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.59296	+	0.467736i	1.12639	+	0.330739i	0.791289	−	0.611442i	\(-0.209410\pi\)
							0.335104	+	0.942181i	\(0.391228\pi\)
\(3\)	−2.26440			−1.30735			−0.653677	−	0.756774i	\(-0.726774\pi\)
							−0.653677	+	0.756774i	\(0.726774\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	−0.226450	+	0.495857i	−0.0855902	+	0.187416i	−0.947587	−	0.319497i	\(-0.896486\pi\)
0.861997	+	0.506913i	\(0.169213\pi\)
\(11\)	2.75599	+	1.84514i	0.830962	+	0.556329i
							\(13\)	−3.43328	−	2.20643i	−0.952220	−	0.611954i	−0.0303852	−	0.999538i	\(-0.509673\pi\)
−0.921834	+	0.387584i	\(0.873310\pi\)
\(17\)	−0.972558	−	6.76429i	−0.235880	−	1.64058i	−0.671898	−	0.740643i	\(-0.734521\pi\)
							0.436018	−	0.899938i	\(-0.356388\pi\)
\(19\)	0.609635	−	4.24010i	0.139860	−	0.972746i	−0.792153	−	0.610322i	\(-0.791040\pi\)
							0.932013	−	0.362424i	\(-0.118051\pi\)
\(23\)	−2.34849	−	5.14247i	−0.489693	−	1.07228i	−0.979683	−	0.200550i	\(-0.935727\pi\)
							0.489990	−	0.871728i	\(-0.337000\pi\)
\(29\)	−0.553978	+	3.85300i	−0.102871	+	0.715484i	0.871477	+	0.490437i	\(0.163163\pi\)
							−0.974348	+	0.225047i	\(0.927746\pi\)
\(31\)	−0.315928	+	0.203034i	−0.0567423	+	0.0364660i	−0.568704	−	0.822542i	\(-0.692555\pi\)
							0.511962	+	0.859008i	\(0.328919\pi\)
\(37\)	−3.29623	+	2.11836i	−0.541896	+	0.348256i	−0.782780	−	0.622298i	\(-0.786199\pi\)
							0.240884	+	0.970554i	\(0.422563\pi\)
\(41\)	−7.40432	−	2.17411i	−1.15636	−	0.339538i	−0.353343	−	0.935494i	\(-0.614955\pi\)
							−0.803017	+	0.595955i	\(0.796773\pi\)
\(43\)	−2.69372	−	3.10872i	−0.410788	−	0.474075i	0.512220	−	0.858854i	\(-0.328823\pi\)
							−0.923009	+	0.384779i	\(0.874278\pi\)
\(47\)	11.8782	−	3.48777i	1.73262	−	0.508743i	0.745197	−	0.666845i	\(-0.232356\pi\)
							0.987423	+	0.158102i	\(0.0505374\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.16	✓	220
121.67	even	11	inner	605.2.k.b.551.16	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.16	✓	220	1.1	even	1	trivial
605.2.k.b.551.16	yes	220	121.67	even	11	inner