Embedded newform 605.2.s.b.16.20

• Label: 605.2.s.b.16.20
• 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			16.20
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28990 + 0.262741i) q^{2} +(1.65951 + 1.20570i) q^{3} +(3.22657 + 0.750306i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(3.48331 + 3.19696i) q^{6} +(-1.38442 - 1.42453i) q^{7} +(2.84961 + 1.01674i) q^{8} +(0.373196 + 1.14858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28990	+	0.262741i	1.61920	+	0.185786i	0.875668	−	0.482913i	\(-0.160421\pi\)
							0.743533	+	0.668699i	\(0.233149\pi\)
\(3\)	1.65951	+	1.20570i	0.958118	+	0.696113i	0.952713	−	0.303872i	\(-0.0982796\pi\)
							0.00540476	+	0.999985i	\(0.498280\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−1.38442	−	1.42453i	−0.523262	−	0.538423i	0.403052	−	0.915177i	\(-0.367949\pi\)
−0.926313	+	0.376754i	\(0.877040\pi\)
\(11\)	1.27706	+	3.06090i	0.385049	+	0.922896i
							\(13\)	0.362552	−	0.601225i	0.100554	−	0.166750i	−0.802004	−	0.597319i	\(-0.796233\pi\)
0.902558	+	0.430569i	\(0.141687\pi\)
\(17\)	−3.78371	+	3.09393i	−0.917685	+	0.750387i	−0.968680	−	0.248312i	\(-0.920124\pi\)
							0.0509952	+	0.998699i	\(0.483761\pi\)
\(19\)	6.59430	+	0.377076i	1.51284	+	0.0865072i	0.793678	−	0.608338i	\(-0.208164\pi\)
							0.719159	+	0.694845i	\(0.244527\pi\)
\(23\)	1.25178	−	8.70635i	0.261015	−	1.81540i	−0.264242	−	0.964456i	\(-0.585122\pi\)
							0.525257	−	0.850944i	\(-0.323969\pi\)
\(29\)	1.36174	−	3.49761i	0.252870	−	0.649490i	−0.747013	−	0.664809i	\(-0.768513\pi\)
							0.999883	+	0.0153192i	\(0.00487644\pi\)
\(31\)	−6.33506	−	2.67722i	−1.13781	−	0.480843i	−0.262754	−	0.964863i	\(-0.584631\pi\)
							−0.875057	+	0.484020i	\(0.839176\pi\)
\(37\)	−0.532051	+	6.19456i	−0.0874687	+	1.01838i	0.812655	+	0.582745i	\(0.198021\pi\)
							−0.900124	+	0.435634i	\(0.856524\pi\)
\(41\)	4.81112	−	2.71696i	0.751370	−	0.424318i	−0.0679247	−	0.997690i	\(-0.521638\pi\)
							0.819295	+	0.573372i	\(0.194365\pi\)
\(43\)	−8.40539	+	2.46804i	−1.28181	+	0.376373i	−0.850569	−	0.525864i	\(-0.823742\pi\)
							−0.431241	+	0.902237i	\(0.641924\pi\)
\(47\)	−0.497648	+	2.45599i	−0.0725894	+	0.358243i	−0.999870	−	0.0161243i	\(-0.994867\pi\)
							0.927281	+	0.374367i	\(0.122140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.20	✓	880
121.53	even	55	inner	605.2.s.b.416.20	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.20	✓	880	1.1	even	1	trivial
605.2.s.b.416.20	yes	880	121.53	even	55	inner