Embedded newform 525.2.bh.a.13.9

• Label: 525.2.bh.a.13.9
• 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			13.9
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60091 + 0.253560i) q^{2} +(-0.891007 - 0.453990i) q^{3} +(0.596516 - 0.193820i) q^{4} +(2.13278 + 0.671755i) q^{5} +(1.54154 + 0.500876i) q^{6} +(-0.369115 - 2.61988i) q^{7} +(1.98258 - 1.01018i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60091	+	0.253560i	−1.13202	+	0.179294i	−0.694208	−	0.719775i	\(-0.744245\pi\)
							−0.437808	+	0.899068i	\(0.644245\pi\)
\(3\)	−0.891007	−	0.453990i	−0.514423	−	0.262112i
							\(5\)	2.13278	+	0.671755i	0.953808	+	0.300418i
\(7\)	−0.369115	−	2.61988i	−0.139512	−	0.990220i
							\(11\)	−1.67350	−	1.21587i	−0.504578	−	0.366598i	0.306185	−	0.951972i	\(-0.400948\pi\)
−0.810763	+	0.585374i	\(0.800948\pi\)
\(13\)	−0.0267962	+	0.169185i	−0.00743194	+	0.0469234i	−0.991126	−	0.132929i	\(-0.957562\pi\)
							0.983694	+	0.179852i	\(0.0575619\pi\)
\(17\)	0.889287	−	0.453114i	0.215684	−	0.109896i	−0.342811	−	0.939404i	\(-0.611379\pi\)
							0.558495	+	0.829508i	\(0.311379\pi\)
\(19\)	−1.18337	+	3.64205i	−0.271484	+	0.835543i	0.718644	+	0.695378i	\(0.244763\pi\)
							−0.990128	+	0.140165i	\(0.955237\pi\)
\(23\)	−0.771163	−	4.86893i	−0.160799	−	1.01524i	−0.927659	−	0.373429i	\(-0.878182\pi\)
							0.766860	−	0.641814i	\(-0.221818\pi\)
\(29\)	8.86059	−	2.87898i	1.64537	−	0.534613i	0.667641	−	0.744484i	\(-0.267304\pi\)
							0.977729	+	0.209870i	\(0.0673042\pi\)
\(31\)	−9.75473	−	3.16950i	−1.75200	−	0.569260i	−0.755679	−	0.654943i	\(-0.772693\pi\)
							−0.996322	+	0.0856830i	\(0.972693\pi\)
\(37\)	0.840593	−	5.30730i	0.138193	−	0.872514i	−0.817023	−	0.576605i	\(-0.804377\pi\)
							0.955216	−	0.295910i	\(-0.0956227\pi\)
\(41\)	−3.90298	−	5.37199i	−0.609542	−	0.838963i	0.386997	−	0.922081i	\(-0.373512\pi\)
							−0.996540	+	0.0831177i	\(0.973512\pi\)
\(43\)	0.374585	+	0.374585i	0.0571237	+	0.0571237i	0.735092	−	0.677968i	\(-0.237139\pi\)
							−0.677968	+	0.735092i	\(0.737139\pi\)
\(47\)	4.58060	−	8.98993i	0.668149	−	1.31132i	−0.269254	−	0.963069i	\(-0.586777\pi\)
							0.937403	−	0.348247i	\(-0.113223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.9	✓	320
7.6	odd	2	inner	525.2.bh.a.13.10	yes	320
25.2	odd	20	inner	525.2.bh.a.202.10	yes	320
175.27	even	20	inner	525.2.bh.a.202.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.9	✓	320	1.1	even	1	trivial
525.2.bh.a.13.10	yes	320	7.6	odd	2	inner
525.2.bh.a.202.9	yes	320	175.27	even	20	inner
525.2.bh.a.202.10	yes	320	25.2	odd	20	inner