Embedded newform 605.2.s.b.16.19

• Label: 605.2.s.b.16.19
• 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			16.19
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13889 + 0.245415i) q^{2} +(-1.59867 - 1.16150i) q^{3} +(2.56661 + 0.596840i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(-3.13432 - 2.87666i) q^{6} +(-2.87657 - 2.95992i) q^{7} +(1.28779 + 0.459482i) q^{8} +(0.279601 + 0.860523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13889	+	0.245415i	1.51243	+	0.173535i	0.830142	−	0.557553i	\(-0.188259\pi\)
							0.682284	+	0.731087i	\(0.260987\pi\)
\(3\)	−1.59867	−	1.16150i	−0.922990	−	0.670591i	0.0212767	−	0.999774i	\(-0.493227\pi\)
							−0.944266	+	0.329182i	\(0.893227\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−2.87657	−	2.95992i	−1.08724	−	1.11874i	−0.992375	−	0.123253i	\(-0.960667\pi\)
−0.0948644	−	0.995490i	\(-0.530242\pi\)
\(11\)	−3.08684	+	1.21301i	−0.930718	+	0.365737i
							\(13\)	2.25345	−	3.73693i	0.624995	−	1.03644i	−0.369253	−	0.929329i	\(-0.620386\pi\)
0.994247	−	0.107108i	\(-0.0341592\pi\)
\(17\)	0.584647	−	0.478064i	0.141798	−	0.115948i	−0.559453	−	0.828862i	\(-0.688989\pi\)
							0.701251	+	0.712914i	\(0.252625\pi\)
\(19\)	−0.538414	−	0.0307876i	−0.123521	−	0.00706317i	−0.00478087	−	0.999989i	\(-0.501522\pi\)
							−0.118740	+	0.992925i	\(0.537885\pi\)
\(23\)	0.613360	−	4.26602i	0.127895	−	0.889526i	−0.820322	−	0.571902i	\(-0.806206\pi\)
							0.948217	−	0.317624i	\(-0.102885\pi\)
\(29\)	0.886648	−	2.27734i	0.164646	−	0.422891i	−0.825235	−	0.564790i	\(-0.808957\pi\)
							0.989881	+	0.141899i	\(0.0453209\pi\)
\(31\)	4.74950	+	2.00716i	0.853036	+	0.360496i	0.771543	−	0.636178i	\(-0.219486\pi\)
							0.0814936	+	0.996674i	\(0.474031\pi\)
\(37\)	−0.267728	+	3.11710i	−0.0440142	+	0.512448i	0.940815	+	0.338921i	\(0.110062\pi\)
							−0.984829	+	0.173527i	\(0.944484\pi\)
\(41\)	−8.80037	+	4.96980i	−1.37439	+	0.776152i	−0.989184	−	0.146679i	\(-0.953141\pi\)
							−0.385203	+	0.922832i	\(0.625869\pi\)
\(43\)	6.67512	−	1.95999i	1.01795	−	0.298896i	0.270146	−	0.962819i	\(-0.412928\pi\)
							0.747801	+	0.663923i	\(0.231110\pi\)
\(47\)	0.180915	−	0.892852i	0.0263892	−	0.130236i	−0.964722	−	0.263270i	\(-0.915199\pi\)
							0.991112	+	0.133034i	\(0.0424719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.19	✓	880
121.53	even	55	inner	605.2.s.b.416.19	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.19	✓	880	1.1	even	1	trivial
605.2.s.b.416.19	yes	880	121.53	even	55	inner