Embedded newform 525.2.bh.a.13.12

• Label: 525.2.bh.a.13.12
• 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			13.12
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56905 + 0.248513i) q^{2} +(0.891007 + 0.453990i) q^{3} +(0.498045 - 0.161825i) q^{4} +(1.85206 - 1.25295i) q^{5} +(-1.51086 - 0.490907i) q^{6} +(-2.52080 - 0.803473i) q^{7} +(2.08968 - 1.06474i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56905	+	0.248513i	−1.10949	+	0.175725i	−0.684172	−	0.729321i	\(-0.739836\pi\)
							−0.425314	+	0.905046i	\(0.639836\pi\)
\(3\)	0.891007	+	0.453990i	0.514423	+	0.262112i
							\(5\)	1.85206	−	1.25295i	0.828266	−	0.560335i
\(7\)	−2.52080	−	0.803473i	−0.952773	−	0.303684i
							\(11\)	−2.10712	−	1.53091i	−0.635321	−	0.461588i	0.222918	−	0.974837i	\(-0.428442\pi\)
−0.858240	+	0.513249i	\(0.828442\pi\)
\(13\)	0.768586	−	4.85266i	0.213167	−	1.34589i	−0.616379	−	0.787449i	\(-0.711401\pi\)
							0.829547	−	0.558437i	\(-0.188599\pi\)
\(17\)	−6.24993	+	3.18450i	−1.51583	+	0.772354i	−0.996609	−	0.0822851i	\(-0.973778\pi\)
							−0.519222	+	0.854639i	\(0.673778\pi\)
\(19\)	1.35198	−	4.16098i	0.310166	−	0.954594i	−0.667532	−	0.744581i	\(-0.732649\pi\)
							0.977699	−	0.210013i	\(-0.0673507\pi\)
\(23\)	−0.291302	−	1.83921i	−0.0607408	−	0.383502i	−0.999267	−	0.0382929i	\(-0.987808\pi\)
							0.938526	−	0.345209i	\(-0.112192\pi\)
\(29\)	−9.03730	+	2.93640i	−1.67818	+	0.545275i	−0.984558	−	0.175059i	\(-0.943988\pi\)
							−0.693627	+	0.720335i	\(0.743988\pi\)
\(31\)	2.23961	+	0.727693i	0.402246	+	0.130698i	0.503151	−	0.864198i	\(-0.332174\pi\)
							−0.100905	+	0.994896i	\(0.532174\pi\)
\(37\)	1.06795	−	6.74275i	0.175569	−	1.10850i	−0.729733	−	0.683733i	\(-0.760355\pi\)
							0.905302	−	0.424769i	\(-0.139645\pi\)
\(41\)	5.68557	+	7.82552i	0.887937	+	1.22214i	0.974159	+	0.225865i	\(0.0725207\pi\)
							−0.0862215	+	0.996276i	\(0.527479\pi\)
\(43\)	−4.59248	−	4.59248i	−0.700347	−	0.700347i	0.264138	−	0.964485i	\(-0.414913\pi\)
							−0.964485	+	0.264138i	\(0.914913\pi\)
\(47\)	6.11140	−	11.9943i	0.891440	−	1.74955i	0.276420	−	0.961037i	\(-0.410852\pi\)
							0.615019	−	0.788512i	\(-0.289148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.12	yes	320
7.6	odd	2	inner	525.2.bh.a.13.11	✓	320
25.2	odd	20	inner	525.2.bh.a.202.11	yes	320
175.27	even	20	inner	525.2.bh.a.202.12	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.11	✓	320	7.6	odd	2	inner
525.2.bh.a.13.12	yes	320	1.1	even	1	trivial
525.2.bh.a.202.11	yes	320	25.2	odd	20	inner
525.2.bh.a.202.12	yes	320	175.27	even	20	inner