Embedded newform 525.2.bo.a.4.5

• Label: 525.2.bo.a.4.5
• Level: $525$
• Weight: $2$
• Character: 525.4
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bo (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	525.4
Dual form			525.2.bo.a.394.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.485930 - 2.28612i) q^{2} +(-0.406737 - 0.913545i) q^{3} +(-3.16312 + 1.40831i) q^{4} +(1.52274 + 1.63746i) q^{5} +(-1.89083 + 1.37377i) q^{6} +(2.64502 - 0.0621505i) q^{7} +(2.00910 + 2.76528i) q^{8} +(-0.669131 + 0.743145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.485930	−	2.28612i	−0.343604	−	1.61653i	−0.722692	−	0.691170i	\(-0.757096\pi\)
							0.379088	−	0.925361i	\(-0.376238\pi\)
\(3\)	−0.406737	−	0.913545i	−0.234830	−	0.527436i
							\(5\)	1.52274	+	1.63746i	0.680990	+	0.732293i
\(7\)	2.64502	−	0.0621505i	0.999724	−	0.0234907i
							\(11\)	3.13526	+	3.48206i	0.945315	+	1.04988i	0.998683	+	0.0513043i	\(0.0163378\pi\)
−0.0533676	+	0.998575i	\(0.516996\pi\)
\(13\)	1.77646	−	0.577205i	0.492700	−	0.160088i	−0.0521202	−	0.998641i	\(-0.516598\pi\)
							0.544820	+	0.838553i	\(0.316598\pi\)
\(17\)	3.16109	−	0.332244i	0.766678	−	0.0805811i	0.286888	−	0.957964i	\(-0.407379\pi\)
							0.479790	+	0.877383i	\(0.340713\pi\)
\(19\)	3.49492	+	1.55604i	0.801790	+	0.356980i	0.766375	−	0.642393i	\(-0.222058\pi\)
							0.0354142	+	0.999373i	\(0.488725\pi\)
\(23\)	−1.25624	−	5.91014i	−0.261944	−	1.23235i	−0.890644	−	0.454701i	\(-0.849746\pi\)
							0.628700	−	0.777648i	\(-0.283587\pi\)
\(29\)	−7.93379	−	5.76424i	−1.47327	−	1.07039i	−0.979650	−	0.200714i	\(-0.935674\pi\)
							−0.493619	−	0.869678i	\(-0.664326\pi\)
\(31\)	0.628424	+	5.97906i	0.112868	+	1.07387i	0.893557	+	0.448950i	\(0.148202\pi\)
							−0.780689	+	0.624920i	\(0.785131\pi\)
\(37\)	3.65169	+	3.28800i	0.600335	+	0.540544i	0.912286	−	0.409554i	\(-0.134316\pi\)
							−0.311951	+	0.950098i	\(0.600983\pi\)
\(41\)	−2.38593	−	7.34315i	−0.372620	−	1.14681i	−0.945070	−	0.326867i	\(-0.894007\pi\)
							0.572450	−	0.819939i	\(-0.305993\pi\)
\(43\)			1.57307i			0.239891i	0.992781	+	0.119945i	\(0.0382719\pi\)
							−0.992781	+	0.119945i	\(0.961728\pi\)
\(47\)	−9.59736	−	1.00872i	−1.39992	−	0.147137i	−0.625704	−	0.780060i	\(-0.715188\pi\)
							−0.774215	+	0.632923i	\(0.781855\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.4.5	✓	320
7.2	even	3	inner	525.2.bo.a.79.5	yes	320
25.19	even	10	inner	525.2.bo.a.319.5	yes	320
175.44	even	30	inner	525.2.bo.a.394.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.5	✓	320	1.1	even	1	trivial
525.2.bo.a.79.5	yes	320	7.2	even	3	inner
525.2.bo.a.319.5	yes	320	25.19	even	10	inner
525.2.bo.a.394.5	yes	320	175.44	even	30	inner