Embedded newform 605.2.s.b.16.17

• Label: 605.2.s.b.16.17
• 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			16.17
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.82551 + 0.209458i) q^{2} +(2.66801 + 1.93842i) q^{3} +(1.34058 + 0.311739i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(4.46445 + 4.09744i) q^{6} +(0.0611421 + 0.0629137i) q^{7} +(-1.07930 - 0.385094i) q^{8} +(2.43374 + 7.49028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.82551	+	0.209458i	1.29083	+	0.148109i	0.732075	−	0.681224i	\(-0.238552\pi\)
							0.558754	+	0.829333i	\(0.311279\pi\)
\(3\)	2.66801	+	1.93842i	1.54038	+	1.11915i	0.950100	+	0.311946i	\(0.100981\pi\)
							0.590275	+	0.807202i	\(0.299019\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	0.0611421	+	0.0629137i	0.0231095	+	0.0237792i	0.728588	−	0.684952i	\(-0.240177\pi\)
−0.705478	+	0.708731i	\(0.749268\pi\)
\(11\)	−1.66821	−	2.86654i	−0.502984	−	0.864295i
							\(13\)	2.98135	−	4.94401i	0.826877	−	1.37122i	−0.0996620	−	0.995021i	\(-0.531776\pi\)
0.926539	−	0.376200i	\(-0.122769\pi\)
\(17\)	2.50624	−	2.04935i	0.607853	−	0.497039i	−0.278261	−	0.960505i	\(-0.589758\pi\)
							0.886115	+	0.463466i	\(0.153394\pi\)
\(19\)	−2.24278	−	0.128247i	−0.514530	−	0.0294219i	−0.202104	−	0.979364i	\(-0.564778\pi\)
							−0.312426	+	0.949942i	\(0.601142\pi\)
\(23\)	−1.19183	+	8.28939i	−0.248515	+	1.72846i	0.358294	+	0.933609i	\(0.383358\pi\)
							−0.606809	+	0.794848i	\(0.707551\pi\)
\(29\)	1.08524	−	2.78741i	0.201524	−	0.517609i	−0.794442	−	0.607339i	\(-0.792237\pi\)
							0.995966	+	0.0897300i	\(0.0286004\pi\)
\(31\)	−5.87266	−	2.48181i	−1.05476	−	0.445746i	−0.208227	−	0.978080i	\(-0.566769\pi\)
							−0.846534	+	0.532334i	\(0.821315\pi\)
\(37\)	−0.512511	+	5.96705i	−0.0842563	+	0.980978i	0.825107	+	0.564976i	\(0.191115\pi\)
							−0.909364	+	0.416002i	\(0.863431\pi\)
\(41\)	2.73780	−	1.54611i	0.427573	−	0.241461i	−0.262895	−	0.964824i	\(-0.584677\pi\)
							0.690468	+	0.723363i	\(0.257405\pi\)
\(43\)	2.87368	−	0.843789i	0.438233	−	0.128677i	−0.0551716	−	0.998477i	\(-0.517571\pi\)
							0.493404	+	0.869800i	\(0.335752\pi\)
\(47\)	0.100800	−	0.497467i	0.0147032	−	0.0725630i	−0.972059	−	0.234736i	\(-0.924577\pi\)
							0.986762	+	0.162173i	\(0.0518502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.17	✓	880
121.53	even	55	inner	605.2.s.b.416.17	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.17	✓	880	1.1	even	1	trivial
605.2.s.b.416.17	yes	880	121.53	even	55	inner