Embedded newform 605.2.s.b.16.3

• Label: 605.2.s.b.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.b.416.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.03821 - 0.233862i) q^{2} +(0.653999 + 0.475158i) q^{3} +(2.15157 + 0.500325i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(-1.22186 - 1.12142i) q^{6} +(-0.0770600 - 0.0792929i) q^{7} +(-0.403791 - 0.144072i) q^{8} +(-0.725112 - 2.23166i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.03821	−	0.233862i	−1.44123	−	0.165366i	−0.642144	−	0.766584i	\(-0.721955\pi\)
							−0.799085	+	0.601218i	\(0.794682\pi\)
\(3\)	0.653999	+	0.475158i	0.377586	+	0.274333i	0.760350	−	0.649514i	\(-0.225028\pi\)
							−0.382763	+	0.923846i	\(0.625028\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−0.0770600	−	0.0792929i	−0.0291260	−	0.0299699i	0.702412	−	0.711770i	\(-0.252106\pi\)
−0.731538	+	0.681800i	\(0.761197\pi\)
\(11\)	−3.07435	+	1.24433i	−0.926952	+	0.375179i
							\(13\)	0.978429	−	1.62254i	0.271367	−	0.450012i	−0.690577	−	0.723259i	\(-0.742643\pi\)
0.961944	+	0.273247i	\(0.0880977\pi\)
\(17\)	−2.66596	+	2.17994i	−0.646589	+	0.528714i	−0.898464	−	0.439046i	\(-0.855316\pi\)
							0.251875	+	0.967760i	\(0.418953\pi\)
\(19\)	−0.367122	−	0.0209928i	−0.0842235	−	0.00481607i	0.0149263	−	0.999889i	\(-0.495249\pi\)
							−0.0991497	+	0.995073i	\(0.531612\pi\)
\(23\)	0.539872	−	3.75489i	0.112571	−	0.782949i	−0.852832	−	0.522186i	\(-0.825117\pi\)
							0.965403	−	0.260763i	\(-0.0839742\pi\)
\(29\)	−0.153525	+	0.394327i	−0.0285090	+	0.0732247i	−0.945399	−	0.325915i	\(-0.894328\pi\)
							0.916890	+	0.399140i	\(0.130691\pi\)
\(31\)	−4.52370	−	1.91173i	−0.812480	−	0.343357i	−0.0568722	−	0.998381i	\(-0.518113\pi\)
							−0.755607	+	0.655025i	\(0.772658\pi\)
\(37\)	0.422389	−	4.91779i	0.0694403	−	0.808479i	−0.875921	−	0.482455i	\(-0.839745\pi\)
							0.945361	−	0.326025i	\(-0.105709\pi\)
\(41\)	−7.16134	+	4.04420i	−1.11841	+	0.631597i	−0.936132	−	0.351649i	\(-0.885621\pi\)
							−0.182282	+	0.983246i	\(0.558348\pi\)
\(43\)	6.37251	−	1.87114i	0.971799	−	0.285346i	0.242964	−	0.970035i	\(-0.421880\pi\)
							0.728835	+	0.684689i	\(0.240062\pi\)
\(47\)	0.457072	−	2.25574i	0.0666708	−	0.329033i	−0.932841	−	0.360289i	\(-0.882678\pi\)
							0.999511	+	0.0312560i	\(0.00995072\pi\)

(See \(a_n\) instead)

Twists

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

    

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