Embedded newform 525.2.bh.a.13.18

• Label: 525.2.bh.a.13.18
• 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			13.18
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.112696 + 0.0178493i) q^{2} +(0.891007 + 0.453990i) q^{3} +(-1.88973 + 0.614011i) q^{4} +(2.18359 - 0.481591i) q^{5} +(-0.108517 - 0.0352592i) q^{6} +(2.31193 + 1.28646i) q^{7} +(0.405335 - 0.206529i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.112696	+	0.0178493i	−0.0796883	+	0.0126214i	−0.196151	−	0.980574i	\(-0.562844\pi\)
							0.116463	+	0.993195i	\(0.462844\pi\)
\(3\)	0.891007	+	0.453990i	0.514423	+	0.262112i
							\(5\)	2.18359	−	0.481591i	0.976532	−	0.215374i
\(7\)	2.31193	+	1.28646i	0.873827	+	0.486237i
							\(11\)	0.822992	+	0.597939i	0.248141	+	0.180285i	0.704903	−	0.709304i	\(-0.250991\pi\)
−0.456761	+	0.889589i	\(0.650991\pi\)
\(13\)	0.938650	−	5.92640i	0.260335	−	1.64369i	−0.417647	−	0.908609i	\(-0.637145\pi\)
							0.677982	−	0.735079i	\(-0.262855\pi\)
\(17\)	−3.30012	+	1.68150i	−0.800398	+	0.407823i	−0.805819	−	0.592162i	\(-0.798274\pi\)
							0.00542113	+	0.999985i	\(0.498274\pi\)
\(19\)	−2.26021	+	6.95621i	−0.518528	+	1.59586i	0.258243	+	0.966080i	\(0.416856\pi\)
							−0.776771	+	0.629784i	\(0.783144\pi\)
\(23\)	0.640059	+	4.04117i	0.133462	+	0.842643i	0.960048	+	0.279834i	\(0.0902794\pi\)
							−0.826587	+	0.562809i	\(0.809721\pi\)
\(29\)	7.26568	−	2.36076i	1.34920	−	0.438383i	0.456779	−	0.889580i	\(-0.349003\pi\)
							0.892424	+	0.451197i	\(0.149003\pi\)
\(31\)	4.53318	+	1.47292i	0.814183	+	0.264544i	0.686368	−	0.727254i	\(-0.259204\pi\)
							0.127814	+	0.991798i	\(0.459204\pi\)
\(37\)	−0.693890	+	4.38105i	−0.114075	+	0.720240i	0.862659	+	0.505786i	\(0.168797\pi\)
							−0.976734	+	0.214454i	\(0.931203\pi\)
\(41\)	−7.00165	−	9.63695i	−1.09347	−	1.50504i	−0.843762	−	0.536718i	\(-0.819664\pi\)
							−0.249712	−	0.968320i	\(-0.580336\pi\)
\(43\)	3.06360	+	3.06360i	0.467195	+	0.467195i	0.901005	−	0.433809i	\(-0.142831\pi\)
							−0.433809	+	0.901005i	\(0.642831\pi\)
\(47\)	1.11811	−	2.19441i	0.163093	−	0.320088i	−0.794968	−	0.606651i	\(-0.792513\pi\)
							0.958062	+	0.286563i	\(0.0925127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.18	yes	320
7.6	odd	2	inner	525.2.bh.a.13.17	✓	320
25.2	odd	20	inner	525.2.bh.a.202.17	yes	320
175.27	even	20	inner	525.2.bh.a.202.18	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.17	✓	320	7.6	odd	2	inner
525.2.bh.a.13.18	yes	320	1.1	even	1	trivial
525.2.bh.a.202.17	yes	320	25.2	odd	20	inner
525.2.bh.a.202.18	yes	320	175.27	even	20	inner