Embedded newform 605.2.s.a.16.7

• Label: 605.2.s.a.16.7
• 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			16.7
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18251 - 0.135680i) q^{2} +(0.544716 + 0.395759i) q^{3} +(-0.568105 - 0.132107i) q^{4} +(0.564443 - 0.825472i) q^{5} +(-0.590435 - 0.541896i) q^{6} +(2.13190 + 2.19368i) q^{7} +(2.89596 + 1.03328i) q^{8} +(-0.786961 - 2.42202i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18251	−	0.135680i	−0.836160	−	0.0959404i	−0.314674	−	0.949200i	\(-0.601895\pi\)
							−0.521487	+	0.853259i	\(0.674622\pi\)
\(3\)	0.544716	+	0.395759i	0.314492	+	0.228492i	0.733821	−	0.679342i	\(-0.237735\pi\)
							−0.419330	+	0.907834i	\(0.637735\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	2.13190	+	2.19368i	0.805784	+	0.829132i	0.988053	−	0.154117i	\(-0.0492532\pi\)
−0.182269	+	0.983249i	\(0.558344\pi\)
\(11\)	−3.07210	−	1.24988i	−0.926273	−	0.376854i
							\(13\)	1.35014	−	2.23895i	0.374460	−	0.620973i	−0.610850	−	0.791746i	\(-0.709172\pi\)
0.985310	+	0.170774i	\(0.0546267\pi\)
\(17\)	1.68175	−	1.37516i	0.407884	−	0.333525i	−0.406709	−	0.913558i	\(-0.633324\pi\)
							0.814593	+	0.580032i	\(0.196960\pi\)
\(19\)	2.50937	+	0.143491i	0.575688	+	0.0329190i	0.342509	−	0.939515i	\(-0.388723\pi\)
							0.233179	+	0.972434i	\(0.425087\pi\)
\(23\)	−1.18325	+	8.22966i	−0.246724	+	1.71600i	0.370177	+	0.928961i	\(0.379297\pi\)
							−0.616901	+	0.787041i	\(0.711612\pi\)
\(29\)	1.71841	−	4.41371i	0.319102	−	0.819606i	−0.677367	−	0.735645i	\(-0.736879\pi\)
							0.996469	−	0.0839610i	\(-0.0267571\pi\)
\(31\)	6.27488	+	2.65179i	1.12700	+	0.476275i	0.871432	−	0.490516i	\(-0.163192\pi\)
							0.255569	+	0.966791i	\(0.417737\pi\)
\(37\)	0.871548	−	10.1472i	0.143282	−	1.66820i	−0.473297	−	0.880903i	\(-0.656936\pi\)
							0.616578	−	0.787294i	\(-0.288518\pi\)
\(41\)	−0.383413	+	0.216524i	−0.0598791	+	0.0338153i	−0.521382	−	0.853324i	\(-0.674583\pi\)
							0.461503	+	0.887139i	\(0.347310\pi\)
\(43\)	4.82517	−	1.41680i	0.735831	−	0.216059i	0.107718	−	0.994182i	\(-0.465646\pi\)
							0.628113	+	0.778122i	\(0.283828\pi\)
\(47\)	2.15302	−	10.6256i	0.314050	−	1.54990i	−0.442987	−	0.896528i	\(-0.646081\pi\)
							0.757038	−	0.653371i	\(-0.226646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.7	✓	880
121.53	even	55	inner	605.2.s.a.416.7	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.7	✓	880	1.1	even	1	trivial
605.2.s.a.416.7	yes	880	121.53	even	55	inner