Embedded newform 525.2.bh.a.328.25

• Label: 525.2.bh.a.328.25
• 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			328.25
Character	\(\chi\)	\(=\)	525.328
Dual form			525.2.bh.a.517.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.416329 + 0.817092i) q^{2} +(-0.987688 + 0.156434i) q^{3} +(0.681261 - 0.937676i) q^{4} +(-1.99858 + 1.00284i) q^{5} +(-0.539025 - 0.741904i) q^{6} +(-1.95603 - 1.78155i) q^{7} +(2.86130 + 0.453186i) q^{8} +(0.951057 - 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.416329	+	0.817092i	0.294389	+	0.577771i	0.990069	−	0.140581i	\(-0.0448969\pi\)
							−0.695680	+	0.718352i	\(0.744897\pi\)
\(3\)	−0.987688	+	0.156434i	−0.570242	+	0.0903175i
							\(5\)	−1.99858	+	1.00284i	−0.893791	+	0.448483i
\(7\)	−1.95603	−	1.78155i	−0.739311	−	0.673364i
							\(11\)	0.0792932	−	0.244039i	0.0239078	−	0.0735806i	−0.938391	−	0.345576i	\(-0.887684\pi\)
0.962299	+	0.271995i	\(0.0876835\pi\)
\(13\)	4.71069	+	2.40022i	1.30651	+	0.665700i	0.961991	−	0.273080i	\(-0.0880422\pi\)
							0.344518	+	0.938780i	\(0.388042\pi\)
\(17\)	7.72002	+	1.22273i	1.87238	+	0.296556i	0.986043	−	0.166488i	\(-0.0532427\pi\)
							0.886335	+	0.463044i	\(0.153243\pi\)
\(19\)	3.91397	−	2.84367i	0.897927	−	0.652382i	−0.0400059	−	0.999199i	\(-0.512738\pi\)
							0.937933	+	0.346817i	\(0.112738\pi\)
\(23\)	1.78113	−	0.907531i	0.371391	−	0.189233i	−0.258326	−	0.966058i	\(-0.583171\pi\)
							0.629717	+	0.776825i	\(0.283171\pi\)
\(29\)	−3.94699	+	5.43256i	−0.732937	+	1.00880i	0.266057	+	0.963957i	\(0.414279\pi\)
							−0.998994	+	0.0448439i	\(0.985721\pi\)
\(31\)	−6.53346	−	8.99253i	−1.17344	−	1.61511i	−0.635363	−	0.772214i	\(-0.719149\pi\)
							−0.538081	−	0.842893i	\(-0.680851\pi\)
\(37\)	2.47713	+	1.26216i	0.407238	+	0.207498i	0.645596	−	0.763679i	\(-0.276609\pi\)
							−0.238358	+	0.971177i	\(0.576609\pi\)
\(41\)	5.64461	−	1.83405i	0.881540	−	0.286430i	0.166944	−	0.985966i	\(-0.446610\pi\)
							0.714597	+	0.699537i	\(0.246610\pi\)
\(43\)	−0.714196	−	0.714196i	−0.108914	−	0.108914i	0.650550	−	0.759464i	\(-0.274539\pi\)
							−0.759464	+	0.650550i	\(0.774539\pi\)
\(47\)	0.700437	+	4.42239i	0.102169	+	0.645072i	0.984626	+	0.174677i	\(0.0558883\pi\)
							−0.882456	+	0.470394i	\(0.844112\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.328.25	✓	320
7.6	odd	2	inner	525.2.bh.a.328.26	yes	320
25.17	odd	20	inner	525.2.bh.a.517.26	yes	320
175.167	even	20	inner	525.2.bh.a.517.25	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.328.25	✓	320	1.1	even	1	trivial
525.2.bh.a.328.26	yes	320	7.6	odd	2	inner
525.2.bh.a.517.25	yes	320	175.167	even	20	inner
525.2.bh.a.517.26	yes	320	25.17	odd	20	inner