Embedded newform 525.2.bh.a.13.15

• Label: 525.2.bh.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.809024 + 0.128137i) q^{2} +(-0.891007 - 0.453990i) q^{3} +(-1.26401 + 0.410703i) q^{4} +(-1.82130 - 1.29725i) q^{5} +(0.779018 + 0.253118i) q^{6} +(2.49866 - 0.869897i) q^{7} +(2.42965 - 1.23797i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.809024	+	0.128137i	−0.572066	+	0.0906064i	−0.435764	−	0.900061i	\(-0.643522\pi\)
							−0.136302	+	0.990667i	\(0.543522\pi\)
\(3\)	−0.891007	−	0.453990i	−0.514423	−	0.262112i
							\(5\)	−1.82130	−	1.29725i	−0.814512	−	0.580147i
\(7\)	2.49866	−	0.869897i	0.944403	−	0.328790i
							\(11\)	0.377160	+	0.274022i	0.113718	+	0.0826209i	0.643191	−	0.765706i	\(-0.277610\pi\)
−0.529473	+	0.848327i	\(0.677610\pi\)
\(13\)	0.440000	−	2.77805i	0.122034	−	0.770492i	−0.848441	−	0.529290i	\(-0.822458\pi\)
							0.970475	−	0.241202i	\(-0.0775416\pi\)
\(17\)	−5.77012	+	2.94002i	−1.39946	+	0.713060i	−0.980788	−	0.195078i	\(-0.937504\pi\)
							−0.418671	+	0.908138i	\(0.637504\pi\)
\(19\)	−0.191284	+	0.588711i	−0.0438835	+	0.135059i	−0.970598	−	0.240707i	\(-0.922621\pi\)
							0.926714	+	0.375767i	\(0.122621\pi\)
\(23\)	−0.508183	−	3.20854i	−0.105963	−	0.669027i	−0.982298	−	0.187324i	\(-0.940018\pi\)
							0.876335	−	0.481703i	\(-0.159982\pi\)
\(29\)	−8.46748	+	2.75125i	−1.57237	+	0.510894i	−0.960077	−	0.279736i	\(-0.909753\pi\)
							−0.612294	+	0.790630i	\(0.709753\pi\)
\(31\)	−9.18838	−	2.98549i	−1.65028	−	0.536209i	−0.671481	−	0.741022i	\(-0.734342\pi\)
							−0.978801	+	0.204812i	\(0.934342\pi\)
\(37\)	−0.953845	+	6.02234i	−0.156811	+	0.990067i	0.776270	+	0.630401i	\(0.217109\pi\)
							−0.933081	+	0.359666i	\(0.882891\pi\)
\(41\)	−2.16513	−	2.98005i	−0.338137	−	0.465405i	0.605759	−	0.795648i	\(-0.292869\pi\)
							−0.943896	+	0.330243i	\(0.892869\pi\)
\(43\)	4.53809	+	4.53809i	0.692052	+	0.692052i	0.962683	−	0.270631i	\(-0.0872323\pi\)
							−0.270631	+	0.962683i	\(0.587232\pi\)
\(47\)	1.29739	−	2.54628i	0.189244	−	0.371413i	−0.776817	−	0.629727i	\(-0.783167\pi\)
							0.966061	+	0.258314i	\(0.0831669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.15	✓	320
7.6	odd	2	inner	525.2.bh.a.13.16	yes	320
25.2	odd	20	inner	525.2.bh.a.202.16	yes	320
175.27	even	20	inner	525.2.bh.a.202.15	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.15	✓	320	1.1	even	1	trivial
525.2.bh.a.13.16	yes	320	7.6	odd	2	inner
525.2.bh.a.202.15	yes	320	175.27	even	20	inner
525.2.bh.a.202.16	yes	320	25.2	odd	20	inner