Embedded newform 525.2.bh.a.328.17

• Label: 525.2.bh.a.328.17
• Level: $525$
• Weight: $2$
• Character: 525.328
• 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			328.17
Character	\(\chi\)	\(=\)	525.328
Dual form			525.2.bh.a.517.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.236019 - 0.463214i) q^{2} +(-0.987688 + 0.156434i) q^{3} +(1.01671 - 1.39938i) q^{4} +(1.07716 - 1.95952i) q^{5} +(0.305576 + 0.420590i) q^{6} +(2.24723 + 1.39641i) q^{7} +(-1.91513 - 0.303327i) q^{8} +(0.951057 - 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{7}{20}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.236019	−	0.463214i	−0.166891	−	0.327542i	0.792381	−	0.610027i	\(-0.208841\pi\)
							−0.959272	+	0.282485i	\(0.908841\pi\)
\(3\)	−0.987688	+	0.156434i	−0.570242	+	0.0903175i
							\(5\)	1.07716	−	1.95952i	0.481721	−	0.876325i
\(7\)	2.24723	+	1.39641i	0.849372	+	0.527794i
							\(11\)	0.894676	−	2.75353i	0.269755	−	0.830220i	−0.720805	−	0.693138i	\(-0.756228\pi\)
0.990560	−	0.137082i	\(-0.0437724\pi\)
\(13\)	5.31717	+	2.70923i	1.47472	+	0.751406i	0.992221	−	0.124492i	\(-0.0397300\pi\)
							0.482497	+	0.875898i	\(0.339730\pi\)
\(17\)	−0.573800	−	0.0908810i	−0.139167	−	0.0220419i	0.0864623	−	0.996255i	\(-0.472444\pi\)
							−0.225629	+	0.974213i	\(0.572444\pi\)
\(19\)	−5.15354	+	3.74426i	−1.18230	+	0.858993i	−0.992430	−	0.122815i	\(-0.960808\pi\)
							−0.189873	+	0.981809i	\(0.560808\pi\)
\(23\)	4.49083	−	2.28819i	0.936404	−	0.477121i	0.0819427	−	0.996637i	\(-0.473888\pi\)
							0.854461	+	0.519516i	\(0.173888\pi\)
\(29\)	−0.540408	+	0.743807i	−0.100351	+	0.138122i	−0.856240	−	0.516579i	\(-0.827205\pi\)
							0.755888	+	0.654701i	\(0.227205\pi\)
\(31\)	−3.38044	−	4.65277i	−0.607144	−	0.835662i	0.389194	−	0.921156i	\(-0.372753\pi\)
							−0.996339	+	0.0854932i	\(0.972753\pi\)
\(37\)	4.09114	+	2.08454i	0.672580	+	0.342696i	0.756703	−	0.653759i	\(-0.226809\pi\)
							−0.0841233	+	0.996455i	\(0.526809\pi\)
\(41\)	−1.35772	+	0.441150i	−0.212040	+	0.0688960i	−0.413111	−	0.910681i	\(-0.635558\pi\)
							0.201071	+	0.979577i	\(0.435558\pi\)
\(43\)	−2.90002	−	2.90002i	−0.442249	−	0.442249i	0.450518	−	0.892767i	\(-0.351239\pi\)
							−0.892767	+	0.450518i	\(0.851239\pi\)
\(47\)	−1.44572	−	9.12789i	−0.210879	−	1.33144i	−0.835060	−	0.550159i	\(-0.814567\pi\)
							0.624180	−	0.781280i	\(-0.285433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.328.17	✓	320
7.6	odd	2	inner	525.2.bh.a.328.18	yes	320
25.17	odd	20	inner	525.2.bh.a.517.18	yes	320
175.167	even	20	inner	525.2.bh.a.517.17	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.328.17	✓	320	1.1	even	1	trivial
525.2.bh.a.328.18	yes	320	7.6	odd	2	inner
525.2.bh.a.517.17	yes	320	175.167	even	20	inner
525.2.bh.a.517.18	yes	320	25.17	odd	20	inner