Embedded newform 605.2.s.a.16.3

• Label: 605.2.s.a.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25925 - 0.259225i) q^{2} +(0.692604 + 0.503206i) q^{3} +(3.08898 + 0.718312i) q^{4} +(0.564443 - 0.825472i) q^{5} +(-1.43432 - 1.31641i) q^{6} +(-1.93384 - 1.98987i) q^{7} +(-2.50894 - 0.895186i) q^{8} +(-0.700567 - 2.15612i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25925	−	0.259225i	−1.59753	−	0.183300i	−0.730983	−	0.682395i	\(-0.760938\pi\)
							−0.866547	+	0.499096i	\(0.833666\pi\)
\(3\)	0.692604	+	0.503206i	0.399875	+	0.290526i	0.769490	−	0.638659i	\(-0.220510\pi\)
							−0.369615	+	0.929185i	\(0.620510\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	−1.93384	−	1.98987i	−0.730922	−	0.752101i	0.245165	−	0.969481i	\(-0.421158\pi\)
−0.976087	+	0.217381i	\(0.930249\pi\)
\(11\)	0.390743	−	3.29353i	0.117813	−	0.993036i
							\(13\)	−1.82409	+	3.02492i	−0.505913	+	0.838962i	−0.999528	−	0.0307182i	\(-0.990221\pi\)
0.493615	+	0.869680i	\(0.335675\pi\)
\(17\)	−2.28353	+	1.86723i	−0.553838	+	0.452871i	−0.868046	−	0.496483i	\(-0.834625\pi\)
							0.314209	+	0.949354i	\(0.398261\pi\)
\(19\)	−5.03718	−	0.288036i	−1.15561	−	0.0660801i	−0.531248	−	0.847217i	\(-0.678277\pi\)
							−0.624360	+	0.781136i	\(0.714640\pi\)
\(23\)	−1.10693	+	7.69887i	−0.230811	+	1.60533i	0.463800	+	0.885940i	\(0.346486\pi\)
							−0.694611	+	0.719385i	\(0.744424\pi\)
\(29\)	−2.18943	+	5.62351i	−0.406567	+	1.04426i	0.568566	+	0.822638i	\(0.307499\pi\)
							−0.975133	+	0.221621i	\(0.928865\pi\)
\(31\)	0.564254	+	0.238456i	0.101343	+	0.0428279i	0.439352	−	0.898315i	\(-0.355208\pi\)
							−0.338009	+	0.941143i	\(0.609753\pi\)
\(37\)	0.609328	−	7.09427i	0.100173	−	1.16629i	−0.757761	−	0.652532i	\(-0.773707\pi\)
							0.857934	−	0.513759i	\(-0.171748\pi\)
\(41\)	3.45083	−	1.94877i	0.538929	−	0.304347i	−0.198130	−	0.980176i	\(-0.563487\pi\)
							0.737059	+	0.675829i	\(0.236214\pi\)
\(43\)	−5.62047	+	1.65032i	−0.857113	+	0.251671i	−0.680625	−	0.732632i	\(-0.738292\pi\)
							−0.176488	+	0.984303i	\(0.556474\pi\)
\(47\)	−1.43546	+	7.08425i	−0.209383	+	1.03334i	0.728142	+	0.685426i	\(0.240384\pi\)
							−0.937525	+	0.347918i	\(0.886889\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.3	✓	880
121.53	even	55	inner	605.2.s.a.416.3	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.3	✓	880	1.1	even	1	trivial
605.2.s.a.416.3	yes	880	121.53	even	55	inner