Embedded newform 605.2.s.a.416.3

• Label: 605.2.s.a.416.3
• Level: $605$
• Weight: $2$
• Character: 605.416
• 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			416.3
Character	\(\chi\)	\(=\)	605.416
Dual form			605.2.s.a.16.3

$q$-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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{53}{55}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.3	✓	880	121.16	even	55	inner
605.2.s.a.416.3	yes	880	1.1	even	1	trivial