Embedded newform 525.2.bh.a.13.13

• Label: 525.2.bh.a.13.13
• Level: $525$
• Weight: $2$
• Character: 525.13
• 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.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.13
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44172 + 0.228346i) q^{2} +(-0.891007 - 0.453990i) q^{3} +(0.124296 - 0.0403862i) q^{4} +(1.71338 + 1.43678i) q^{5} +(1.38825 + 0.451069i) q^{6} +(1.34100 + 2.28073i) q^{7} +(2.43121 - 1.23876i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44172	+	0.228346i	−1.01945	+	0.161465i	−0.643717	−	0.765264i	\(-0.722609\pi\)
							−0.375732	+	0.926729i	\(0.622609\pi\)
\(3\)	−0.891007	−	0.453990i	−0.514423	−	0.262112i
							\(5\)	1.71338	+	1.43678i	0.766245	+	0.642548i
\(7\)	1.34100	+	2.28073i	0.506850	+	0.862034i
							\(11\)	0.336688	+	0.244618i	0.101515	+	0.0737551i	0.637385	−	0.770546i	\(-0.280016\pi\)
−0.535870	+	0.844301i	\(0.680016\pi\)
\(13\)	0.293555	−	1.85344i	0.0814176	−	0.514051i	−0.912951	−	0.408070i	\(-0.866202\pi\)
							0.994368	−	0.105981i	\(-0.0337981\pi\)
\(17\)	−4.14136	+	2.11013i	−1.00443	+	0.511781i	−0.877217	−	0.480094i	\(-0.840602\pi\)
							−0.127211	+	0.991876i	\(0.540602\pi\)
\(19\)	2.57638	−	7.92929i	0.591063	−	1.81910i	0.0176392	−	0.999844i	\(-0.494385\pi\)
							0.573423	−	0.819259i	\(-0.305615\pi\)
\(23\)	1.30750	+	8.25521i	0.272632	+	1.72133i	0.620861	+	0.783921i	\(0.286783\pi\)
							−0.348229	+	0.937409i	\(0.613217\pi\)
\(29\)	−0.248653	+	0.0807922i	−0.0461737	+	0.0150027i	−0.332013	−	0.943275i	\(-0.607728\pi\)
							0.285839	+	0.958278i	\(0.407728\pi\)
\(31\)	5.95543	+	1.93504i	1.06963	+	0.347543i	0.790342	−	0.612666i	\(-0.209903\pi\)
							0.279284	+	0.960209i	\(0.409903\pi\)
\(37\)	−0.140783	+	0.888869i	−0.0231446	+	0.146129i	−0.996554	−	0.0829413i	\(-0.973569\pi\)
							0.973410	+	0.229070i	\(0.0735686\pi\)
\(41\)	2.97536	+	4.09523i	0.464673	+	0.639567i	0.975470	−	0.220134i	\(-0.0706496\pi\)
							−0.510797	+	0.859701i	\(0.670650\pi\)
\(43\)	2.51007	+	2.51007i	0.382783	+	0.382783i	0.872104	−	0.489321i	\(-0.162755\pi\)
							−0.489321	+	0.872104i	\(0.662755\pi\)
\(47\)	−2.12998	+	4.18032i	−0.310689	+	0.609762i	−0.992566	−	0.121705i	\(-0.961164\pi\)
							0.681877	+	0.731467i	\(0.261164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.13	✓	320
7.6	odd	2	inner	525.2.bh.a.13.14	yes	320
25.2	odd	20	inner	525.2.bh.a.202.14	yes	320
175.27	even	20	inner	525.2.bh.a.202.13	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.13	✓	320	1.1	even	1	trivial
525.2.bh.a.13.14	yes	320	7.6	odd	2	inner
525.2.bh.a.202.13	yes	320	175.27	even	20	inner
525.2.bh.a.202.14	yes	320	25.2	odd	20	inner