Embedded newform 525.2.bk.a.8.4

• Label: 525.2.bk.a.8.4
• Level: $525$
• Weight: $2$
• Character: 525.8
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			8.4
Character	\(\chi\)	\(=\)	525.8
Dual form			525.2.bk.a.197.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14547 + 2.24812i) q^{2} +(-1.38991 + 1.03351i) q^{3} +(-2.56637 - 3.53230i) q^{4} +(1.60960 - 1.55216i) q^{5} +(-0.731341 - 4.30855i) q^{6} +(0.707107 - 0.707107i) q^{7} +(5.89662 - 0.933933i) q^{8} +(0.863721 - 2.87298i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{3}{20}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14547	+	2.24812i	−0.809973	+	1.58966i	−0.00232467	+	0.999997i	\(0.500740\pi\)
							−0.807648	+	0.589664i	\(0.799260\pi\)
\(3\)	−1.38991	+	1.03351i	−0.802467	+	0.596696i
							\(5\)	1.60960	−	1.55216i	0.719833	−	0.694147i
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	0.833130	−	0.270700i	0.251198	−	0.0816192i	−0.180711	−	0.983536i	\(-0.557840\pi\)
0.431910	+	0.901917i	\(0.357840\pi\)
\(13\)	−4.92619	+	2.51002i	−1.36628	+	0.696154i	−0.974601	−	0.223948i	\(-0.928105\pi\)
							−0.391678	+	0.920102i	\(0.628105\pi\)
\(17\)	0.935084	+	5.90389i	0.226791	+	1.43190i	0.793794	+	0.608187i	\(0.208103\pi\)
							−0.567003	+	0.823716i	\(0.691897\pi\)
\(19\)	−1.60878	+	2.21430i	−0.369080	+	0.507995i	−0.952651	−	0.304067i	\(-0.901655\pi\)
							0.583571	+	0.812062i	\(0.301655\pi\)
\(23\)	5.21108	+	2.65518i	1.08659	+	0.553643i	0.903122	−	0.429383i	\(-0.141269\pi\)
							0.183464	+	0.983026i	\(0.441269\pi\)
\(29\)	−7.97915	+	5.79719i	−1.48169	+	1.07651i	−0.504684	+	0.863304i	\(0.668391\pi\)
							−0.977007	+	0.213208i	\(0.931609\pi\)
\(31\)	4.94766	+	3.59469i	0.888627	+	0.645625i	0.935520	−	0.353275i	\(-0.114932\pi\)
							−0.0468930	+	0.998900i	\(0.514932\pi\)
\(37\)	0.375245	+	0.736459i	0.0616898	+	0.121073i	0.919793	−	0.392405i	\(-0.128357\pi\)
							−0.858103	+	0.513478i	\(0.828357\pi\)
\(41\)	8.27343	+	2.68820i	1.29209	+	0.419826i	0.872824	−	0.488035i	\(-0.162286\pi\)
							0.419269	+	0.907862i	\(0.362286\pi\)
\(43\)	−1.84452	−	1.84452i	−0.281286	−	0.281286i	0.552336	−	0.833622i	\(-0.313737\pi\)
							−0.833622	+	0.552336i	\(0.813737\pi\)
\(47\)	8.37043	+	1.32575i	1.22095	+	0.193380i	0.733451	−	0.679742i	\(-0.237908\pi\)
							0.487501	+	0.873122i	\(0.337908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bk.a.8.4	✓	480
3.2	odd	2	inner	525.2.bk.a.8.57	yes	480
25.22	odd	20	inner	525.2.bk.a.197.57	yes	480
75.47	even	20	inner	525.2.bk.a.197.4	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bk.a.8.4	✓	480	1.1	even	1	trivial
525.2.bk.a.8.57	yes	480	3.2	odd	2	inner
525.2.bk.a.197.4	yes	480	75.47	even	20	inner
525.2.bk.a.197.57	yes	480	25.22	odd	20	inner