Embedded newform 605.2.s.b.16.1

• Label: 605.2.s.b.16.1
• 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			16.1
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61070 - 0.299550i) q^{2} +(-0.194849 - 0.141566i) q^{3} +(4.77802 + 1.11108i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(0.466287 + 0.427954i) q^{6} +(3.66250 + 3.76862i) q^{7} +(-7.19116 - 2.56580i) q^{8} +(-0.909126 - 2.79800i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61070	−	0.299550i	−1.84605	−	0.211814i	−0.880860	−	0.473377i	\(-0.843035\pi\)
							−0.965187	+	0.261563i	\(0.915762\pi\)
\(3\)	−0.194849	−	0.141566i	−0.112496	−	0.0817332i	0.530115	−	0.847926i	\(-0.322149\pi\)
							−0.642611	+	0.766193i	\(0.722149\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	3.66250	+	3.76862i	1.38430	+	1.42441i	0.776081	+	0.630634i	\(0.217205\pi\)
0.608215	+	0.793773i	\(0.291886\pi\)
\(11\)	0.688990	−	3.24427i	0.207738	−	0.978184i
							\(13\)	1.31090	−	2.17389i	0.363580	−	0.602929i	−0.619742	−	0.784806i	\(-0.712763\pi\)
0.983321	+	0.181877i	\(0.0582172\pi\)
\(17\)	1.75348	−	1.43381i	0.425282	−	0.347751i	−0.396023	−	0.918240i	\(-0.629610\pi\)
							0.821305	+	0.570489i	\(0.193246\pi\)
\(19\)	1.34916	+	0.0771475i	0.309518	+	0.0176989i	0.211158	−	0.977452i	\(-0.432277\pi\)
							0.0983600	+	0.995151i	\(0.468640\pi\)
\(23\)	−0.589687	+	4.10136i	−0.122958	+	0.855193i	0.831218	+	0.555947i	\(0.187644\pi\)
							−0.954176	+	0.299246i	\(0.903265\pi\)
\(29\)	1.33714	−	3.43442i	0.248301	−	0.637755i	−0.751457	−	0.659782i	\(-0.770648\pi\)
							0.999757	+	0.0220274i	\(0.00701212\pi\)
\(31\)	−3.62298	−	1.53109i	−0.650707	−	0.274991i	0.0388252	−	0.999246i	\(-0.487638\pi\)
							−0.689532	+	0.724255i	\(0.742184\pi\)
\(37\)	0.404551	−	4.71010i	0.0665077	−	0.774335i	−0.884776	−	0.466017i	\(-0.845688\pi\)
							0.951283	−	0.308318i	\(-0.0997661\pi\)
\(41\)	8.68819	−	4.90645i	1.35687	−	0.766258i	0.370191	−	0.928955i	\(-0.379292\pi\)
							0.986676	+	0.162697i	\(0.0520194\pi\)
\(43\)	−0.132425	+	0.0388836i	−0.0201947	+	0.00592969i	−0.291814	−	0.956475i	\(-0.594259\pi\)
							0.271620	+	0.962405i	\(0.412441\pi\)
\(47\)	−2.45027	+	12.0925i	−0.357408	+	1.76388i	0.248783	+	0.968559i	\(0.419970\pi\)
							−0.606191	+	0.795319i	\(0.707303\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.1	✓	880
121.53	even	55	inner	605.2.s.b.416.1	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.1	✓	880	1.1	even	1	trivial
605.2.s.b.416.1	yes	880	121.53	even	55	inner