Embedded newform 605.2.s.b.16.7

• Label: 605.2.s.b.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.b.416.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42148 - 0.163100i) q^{2} +(1.67703 + 1.21843i) q^{3} +(0.0459846 + 0.0106933i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(-2.18514 - 2.00550i) q^{6} +(2.67320 + 2.75065i) q^{7} +(2.63157 + 0.938944i) q^{8} +(0.400791 + 1.23351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42148	−	0.163100i	−1.00514	−	0.115329i	−0.404239	−	0.914653i	\(-0.632464\pi\)
							−0.600900	+	0.799324i	\(0.705191\pi\)
\(3\)	1.67703	+	1.21843i	0.968231	+	0.703461i	0.955048	−	0.296452i	\(-0.0958035\pi\)
							0.0131833	+	0.999913i	\(0.495803\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	2.67320	+	2.75065i	1.01037	+	1.03965i	0.999211	+	0.0397114i	\(0.0126439\pi\)
0.0111617	+	0.999938i	\(0.496447\pi\)
\(11\)	2.07924	+	2.58394i	0.626916	+	0.779087i
							\(13\)	0.816162	−	1.35345i	0.226363	−	0.375380i	−0.722346	−	0.691532i	\(-0.756936\pi\)
0.948709	+	0.316151i	\(0.102391\pi\)
\(17\)	2.68996	−	2.19957i	0.652411	−	0.533474i	−0.247864	−	0.968795i	\(-0.579729\pi\)
							0.900275	+	0.435321i	\(0.143365\pi\)
\(19\)	−6.96949	−	0.398530i	−1.59891	−	0.0914290i	−0.765260	−	0.643721i	\(-0.777390\pi\)
							−0.833651	+	0.552292i	\(0.813753\pi\)
\(23\)	−0.200135	+	1.39197i	−0.0417309	+	0.290245i	0.958260	+	0.285899i	\(0.0922923\pi\)
							−0.999991	+	0.00434559i	\(0.998617\pi\)
\(29\)	−1.34173	+	3.44620i	−0.249153	+	0.639944i	−0.999784	−	0.0207775i	\(-0.993386\pi\)
							0.750631	+	0.660721i	\(0.229749\pi\)
\(31\)	8.55575	+	3.61569i	1.53666	+	0.649397i	0.983043	−	0.183375i	\(-0.0587022\pi\)
							0.553616	+	0.832772i	\(0.313248\pi\)
\(37\)	−0.827834	+	9.63829i	−0.136095	+	1.58453i	0.536102	+	0.844153i	\(0.319896\pi\)
							−0.672197	+	0.740372i	\(0.734649\pi\)
\(41\)	5.51235	−	3.11297i	0.860884	−	0.486164i	0.00299335	−	0.999996i	\(-0.499047\pi\)
							0.857891	+	0.513832i	\(0.171774\pi\)
\(43\)	−12.3054	+	3.61319i	−1.87656	+	0.551007i	−0.879362	+	0.476154i	\(0.842031\pi\)
							−0.997195	+	0.0748530i	\(0.976151\pi\)
\(47\)	0.181381	−	0.895150i	0.0264571	−	0.130571i	−0.964677	−	0.263435i	\(-0.915145\pi\)
							0.991134	+	0.132864i	\(0.0424173\pi\)

(See \(a_n\) instead)

Twists

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

    

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