Embedded newform 605.2.u.a.4.2

• Label: 605.2.u.a.4.2
• Level: $605$
• Weight: $2$
• Character: 605.4
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $2560$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.u (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(2560\)
Relative dimension:	\(64\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.158138 + 2.76551i) q^{2} +(-1.72481 - 0.560425i) q^{3} +(-5.63610 - 0.646682i) q^{4} +(2.21911 - 0.274891i) q^{5} +(1.82262 - 4.68137i) q^{6} +(0.760205 - 1.79886i) q^{7} +(1.73499 - 10.0255i) q^{8} +(0.233849 + 0.169902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.158138	+	2.76551i	−0.111820	+	1.95551i	0.145911	+	0.989298i	\(0.453389\pi\)
							−0.257731	+	0.966217i	\(0.582975\pi\)
\(3\)	−1.72481	−	0.560425i	−0.995821	−	0.323562i	−0.234626	−	0.972086i	\(-0.575387\pi\)
							−0.761194	+	0.648524i	\(0.775387\pi\)
\(5\)	2.21911	−	0.274891i	0.992415	−	0.122935i
							\(7\)	0.760205	−	1.79886i	0.287330	−	0.679905i	−0.712431	−	0.701742i	\(-0.752406\pi\)
0.999762	+	0.0218367i	\(0.00695140\pi\)
\(11\)	−0.244136	+	3.30763i	−0.0736097	+	0.997287i
							\(13\)	0.0318336	+	0.0563699i	0.00882904	+	0.0156342i	0.875126	−	0.483896i	\(-0.160779\pi\)
−0.866296	+	0.499530i	\(0.833506\pi\)
\(17\)	0.608951	+	1.70670i	0.147692	+	0.413937i	0.992669	−	0.120867i	\(-0.0385674\pi\)
							−0.844976	+	0.534804i	\(0.820386\pi\)
\(19\)	−0.191656	+	6.70884i	−0.0439689	+	1.53911i	0.615956	+	0.787781i	\(0.288770\pi\)
							−0.659925	+	0.751332i	\(0.729412\pi\)
\(23\)	2.41122	−	2.08933i	0.502774	−	0.435656i	−0.366184	−	0.930543i	\(-0.619336\pi\)
							0.868958	+	0.494887i	\(0.164790\pi\)
\(29\)	2.05698	+	3.00823i	0.381971	+	0.558615i	0.967768	−	0.251843i	\(-0.0810366\pi\)
							−0.585797	+	0.810458i	\(0.699218\pi\)
\(31\)	−0.242631	+	1.19743i	−0.0435779	+	0.215065i	−0.995955	−	0.0898589i	\(-0.971358\pi\)
							0.952377	+	0.304924i	\(0.0986311\pi\)
\(37\)	2.12797	+	2.31858i	0.349836	+	0.381172i	0.886581	−	0.462573i	\(-0.153074\pi\)
							−0.536745	+	0.843745i	\(0.680346\pi\)
\(41\)	−1.47603	−	5.61542i	−0.230518	−	0.876981i	−0.977291	−	0.211902i	\(-0.932034\pi\)
							0.746773	−	0.665079i	\(-0.231602\pi\)
\(43\)	5.79522	−	0.833227i	0.883763	−	0.127066i	0.314535	−	0.949246i	\(-0.398151\pi\)
							0.569228	+	0.822180i	\(0.307242\pi\)
\(47\)	−7.28228	−	8.90585i	−1.06223	−	1.29905i	−0.952317	−	0.305111i	\(-0.901307\pi\)
							−0.109913	−	0.993941i	\(-0.535057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.2	✓	2560
5.4	even	2	inner	605.2.u.a.4.63	yes	2560
121.91	even	55	inner	605.2.u.a.454.63	yes	2560
605.454	even	110	inner	605.2.u.a.454.2	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.2	✓	2560	1.1	even	1	trivial
605.2.u.a.4.63	yes	2560	5.4	even	2	inner
605.2.u.a.454.2	yes	2560	605.454	even	110	inner
605.2.u.a.454.63	yes	2560	121.91	even	55	inner