Embedded newform 525.2.bo.a.319.35

• Label: 525.2.bo.a.319.35
• Level: $525$
• Weight: $2$
• Character: 525.319
• 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			319.35
Character	\(\chi\)	\(=\)	525.319
Dual form			525.2.bo.a.79.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.56627 + 1.41028i) q^{2} +(-0.994522 + 0.104528i) q^{3} +(0.255269 + 2.42872i) q^{4} +(-0.0681074 - 2.23503i) q^{5} +(-1.70511 - 1.23883i) q^{6} +(2.54248 + 0.731967i) q^{7} +(-0.547687 + 0.753826i) q^{8} +(0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.56627	+	1.41028i	1.10752	+	0.997218i	0.999995	+	0.00301561i	\(0.000959900\pi\)
							0.107527	+	0.994202i	\(0.465707\pi\)
\(3\)	−0.994522	+	0.104528i	−0.574187	+	0.0603495i
							\(5\)	−0.0681074	−	2.23503i	−0.0304585	−	0.999536i
\(7\)	2.54248	+	0.731967i	0.960969	+	0.276658i
							\(11\)	4.27410	+	0.908489i	1.28869	+	0.273920i	0.800772	−	0.598969i	\(-0.204423\pi\)
0.487919	+	0.872889i	\(0.337756\pi\)
\(13\)	−3.24064	−	1.05295i	−0.898791	−	0.292035i	−0.177053	−	0.984201i	\(-0.556656\pi\)
							−0.721738	+	0.692166i	\(0.756656\pi\)
\(17\)	0.693059	+	1.55664i	0.168092	+	0.377540i	0.977875	−	0.209189i	\(-0.0670823\pi\)
							−0.809784	+	0.586728i	\(0.800416\pi\)
\(19\)	−0.0799488	+	0.760662i	−0.0183415	+	0.174508i	−0.999858	−	0.0168714i	\(-0.994629\pi\)
							0.981516	+	0.191379i	\(0.0612961\pi\)
\(23\)	1.82173	+	1.64029i	0.379857	+	0.342024i	0.836837	−	0.547453i	\(-0.184402\pi\)
							−0.456980	+	0.889477i	\(0.651069\pi\)
\(29\)	−0.565629	+	0.410953i	−0.105035	+	0.0763121i	−0.639063	−	0.769155i	\(-0.720678\pi\)
							0.534028	+	0.845467i	\(0.320678\pi\)
\(31\)	−1.76147	+	0.784259i	−0.316370	+	0.140857i	−0.558778	−	0.829317i	\(-0.688730\pi\)
							0.242408	+	0.970174i	\(0.422063\pi\)
\(37\)	0.244306	+	1.14937i	0.0401637	+	0.188955i	0.993656	−	0.112462i	\(-0.0358737\pi\)
							−0.953492	+	0.301417i	\(0.902540\pi\)
\(41\)	−2.64827	+	8.15052i	−0.413590	+	1.27290i	0.499917	+	0.866073i	\(0.333364\pi\)
							−0.913506	+	0.406824i	\(0.866636\pi\)
\(43\)		−	13.0204i		−	1.98559i	−0.119846	−	0.992793i	\(-0.538240\pi\)
							0.119846	−	0.992793i	\(-0.461760\pi\)
\(47\)	1.98879	−	4.46690i	0.290095	−	0.651565i	−0.708434	−	0.705777i	\(-0.750598\pi\)
							0.998529	+	0.0542122i	\(0.0172647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.319.35	yes	320
7.2	even	3	inner	525.2.bo.a.394.35	yes	320
25.4	even	10	inner	525.2.bo.a.4.35	✓	320
175.79	even	30	inner	525.2.bo.a.79.35	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.35	✓	320	25.4	even	10	inner
525.2.bo.a.79.35	yes	320	175.79	even	30	inner
525.2.bo.a.319.35	yes	320	1.1	even	1	trivial
525.2.bo.a.394.35	yes	320	7.2	even	3	inner