Embedded newform 605.2.s.b.16.5

• Label: 605.2.s.b.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72317 - 0.197715i) q^{2} +(-2.49016 - 1.80921i) q^{3} +(0.982201 + 0.228401i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(3.93327 + 3.60992i) q^{6} +(-0.450743 - 0.463803i) q^{7} +(1.61987 + 0.577969i) q^{8} +(2.00063 + 6.15729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72317	−	0.197715i	−1.21847	−	0.139806i	−0.519190	−	0.854659i	\(-0.673766\pi\)
							−0.699275	+	0.714853i	\(0.746494\pi\)
\(3\)	−2.49016	−	1.80921i	−1.43770	−	1.04455i	−0.988517	−	0.151107i	\(-0.951716\pi\)
							−0.449180	−	0.893441i	\(-0.648284\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−0.450743	−	0.463803i	−0.170365	−	0.175301i	0.626573	−	0.779363i	\(-0.284457\pi\)
−0.796938	+	0.604062i	\(0.793548\pi\)
\(11\)	−3.30875	+	0.228440i	−0.997625	+	0.0688772i
							\(13\)	3.16457	−	5.24785i	0.877693	−	1.45549i	−0.0125077	−	0.999922i	\(-0.503981\pi\)
0.890201	−	0.455569i	\(-0.150564\pi\)
\(17\)	5.35056	−	4.37513i	1.29770	−	1.06112i	0.303584	−	0.952805i	\(-0.401817\pi\)
							0.994116	−	0.108320i	\(-0.0345470\pi\)
\(19\)	6.34879	+	0.363037i	1.45651	+	0.0832864i	0.767308	−	0.641279i	\(-0.221596\pi\)
							0.689206	+	0.724566i	\(0.257960\pi\)
\(23\)	−0.520462	+	3.61989i	−0.108524	+	0.754800i	0.860788	+	0.508964i	\(0.169972\pi\)
							−0.969311	+	0.245836i	\(0.920938\pi\)
\(29\)	−1.60283	+	4.11683i	−0.297638	+	0.764476i	0.701043	+	0.713119i	\(0.252718\pi\)
							−0.998680	+	0.0513568i	\(0.983645\pi\)
\(31\)	−2.52903	−	1.06878i	−0.454228	−	0.191958i	0.150138	−	0.988665i	\(-0.452028\pi\)
							−0.604366	+	0.796707i	\(0.706574\pi\)
\(37\)	0.144073	−	1.67741i	0.0236854	−	0.275765i	−0.974902	−	0.222634i	\(-0.928535\pi\)
							0.998588	−	0.0531307i	\(-0.0169200\pi\)
\(41\)	7.86724	−	4.44284i	1.22866	−	0.693855i	0.265842	−	0.964017i	\(-0.414350\pi\)
							0.962815	+	0.270162i	\(0.0870772\pi\)
\(43\)	−11.4490	+	3.36173i	−1.74596	+	0.512659i	−0.989890	−	0.141836i	\(-0.954700\pi\)
							−0.756066	+	0.654495i	\(0.772881\pi\)
\(47\)	0.567507	−	2.80076i	0.0827794	−	0.408533i	−0.917170	−	0.398496i	\(-0.869532\pi\)
							0.999950	−	0.0100369i	\(-0.00319488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.5	✓	880
121.53	even	55	inner	605.2.s.b.416.5	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.5	✓	880	1.1	even	1	trivial
605.2.s.b.416.5	yes	880	121.53	even	55	inner