Embedded newform 605.2.u.a.4.17

• Label: 605.2.u.a.4.17
• 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			4.17
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0872065 + 1.52507i) q^{2} +(-0.0606761 - 0.0197148i) q^{3} +(-0.331261 - 0.0380087i) q^{4} +(-2.10361 - 0.758176i) q^{5} +(0.0353578 - 0.0908158i) q^{6} +(-0.188162 + 0.445244i) q^{7} +(-0.434111 + 2.50849i) q^{8} +(-2.42376 - 1.76096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0872065	+	1.52507i	−0.0616643	+	1.07839i	0.809413	+	0.587240i	\(0.199785\pi\)
							−0.871077	+	0.491146i	\(0.836578\pi\)
\(3\)	−0.0606761	−	0.0197148i	−0.0350313	−	0.0113824i	0.291449	−	0.956586i	\(-0.405863\pi\)
							−0.326480	+	0.945204i	\(0.605863\pi\)
\(5\)	−2.10361	−	0.758176i	−0.940762	−	0.339066i
							\(7\)	−0.188162	+	0.445244i	−0.0711185	+	0.168286i	−0.952832	−	0.303499i	\(-0.901845\pi\)
0.881713	+	0.471786i	\(0.156390\pi\)
\(11\)	0.565685	−	3.26803i	0.170560	−	0.985347i
							\(13\)	−2.46784	−	4.36998i	−0.684455	−	1.21201i	−0.967491	−	0.252904i	\(-0.918614\pi\)
0.283036	−	0.959109i	\(-0.408658\pi\)
\(17\)	−0.884142	−	2.47798i	−0.214436	−	0.600999i	0.785402	−	0.618986i	\(-0.212456\pi\)
							−0.999838	+	0.0179870i	\(0.994274\pi\)
\(19\)	−0.115951	+	4.05880i	−0.0266009	+	0.931153i	0.871246	+	0.490846i	\(0.163312\pi\)
							−0.897847	+	0.440307i	\(0.854870\pi\)
\(23\)	−1.35868	+	1.17731i	−0.283305	+	0.245485i	−0.784907	−	0.619613i	\(-0.787290\pi\)
							0.501602	+	0.865098i	\(0.332744\pi\)
\(29\)	−4.51188	−	6.59841i	−0.837835	−	1.22529i	−0.972151	−	0.234356i	\(-0.924702\pi\)
							0.134316	−	0.990939i	\(-0.457116\pi\)
\(31\)	0.200242	−	0.988232i	0.0359645	−	0.177492i	−0.958074	−	0.286521i	\(-0.907501\pi\)
							0.994039	+	0.109029i	\(0.0347741\pi\)
\(37\)	−1.57374	−	1.71470i	−0.258721	−	0.281896i	0.593772	−	0.804633i	\(-0.297638\pi\)
							−0.852494	+	0.522738i	\(0.824911\pi\)
\(41\)	−2.29885	−	8.74573i	−0.359020	−	1.36585i	−0.864334	−	0.502918i	\(-0.832260\pi\)
							0.505314	−	0.862936i	\(-0.331377\pi\)
\(43\)	−0.128223	+	0.0184357i	−0.0195539	+	0.00281142i	−0.152085	−	0.988367i	\(-0.548599\pi\)
							0.132531	+	0.991179i	\(0.457690\pi\)
\(47\)	−2.98931	−	3.65577i	−0.436036	−	0.533249i	0.508760	−	0.860908i	\(-0.330104\pi\)
							−0.944796	+	0.327659i	\(0.893740\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.17	✓	2560
5.4	even	2	inner	605.2.u.a.4.48	yes	2560
121.91	even	55	inner	605.2.u.a.454.48	yes	2560
605.454	even	110	inner	605.2.u.a.454.17	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.17	✓	2560	1.1	even	1	trivial
605.2.u.a.4.48	yes	2560	5.4	even	2	inner
605.2.u.a.454.17	yes	2560	605.454	even	110	inner
605.2.u.a.454.48	yes	2560	121.91	even	55	inner