Embedded newform 525.2.bk.a.8.18

• Label: 525.2.bk.a.8.18
• 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			8.18
Character	\(\chi\)	\(=\)	525.8
Dual form			525.2.bk.a.197.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.640516 + 1.25708i) q^{2} +(-1.19705 - 1.25183i) q^{3} +(0.00557249 + 0.00766987i) q^{4} +(2.19052 - 0.449020i) q^{5} +(2.34038 - 0.702977i) q^{6} +(0.707107 - 0.707107i) q^{7} +(-2.80019 + 0.443506i) q^{8} +(-0.134142 + 2.99700i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.640516	+	1.25708i	−0.452913	+	0.888892i	0.545787	+	0.837924i	\(0.316231\pi\)
							−0.998700	+	0.0509683i	\(0.983769\pi\)
\(3\)	−1.19705	−	1.25183i	−0.691117	−	0.722743i
							\(5\)	2.19052	−	0.449020i	0.979631	−	0.200808i
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	5.34515	−	1.73674i	1.61162	−	0.523648i	0.641678	−	0.766974i	\(-0.278239\pi\)
0.969944	+	0.243327i	\(0.0782387\pi\)
\(13\)	−1.94868	+	0.992903i	−0.540467	+	0.275382i	−0.702840	−	0.711348i	\(-0.748085\pi\)
							0.162373	+	0.986729i	\(0.448085\pi\)
\(17\)	−1.19775	−	7.56231i	−0.290497	−	1.83413i	−0.512020	−	0.858974i	\(-0.671103\pi\)
							0.221522	−	0.975155i	\(-0.428897\pi\)
\(19\)	−2.89923	+	3.99044i	−0.665128	+	0.915470i	−0.999638	−	0.0269221i	\(-0.991429\pi\)
							0.334509	+	0.942392i	\(0.391429\pi\)
\(23\)	6.27539	+	3.19747i	1.30851	+	0.666718i	0.962441	−	0.271491i	\(-0.0875169\pi\)
							0.346067	+	0.938210i	\(0.387517\pi\)
\(29\)	5.46037	−	3.96719i	1.01396	−	0.736688i	0.0489275	−	0.998802i	\(-0.484420\pi\)
							0.965037	+	0.262114i	\(0.0844197\pi\)
\(31\)	−1.01399	−	0.736707i	−0.182118	−	0.132316i	0.492991	−	0.870034i	\(-0.335903\pi\)
							−0.675109	+	0.737718i	\(0.735903\pi\)
\(37\)	3.09253	+	6.06942i	0.508408	+	0.997807i	0.992437	+	0.122755i	\(0.0391730\pi\)
							−0.484029	+	0.875052i	\(0.660827\pi\)
\(41\)	5.61395	+	1.82408i	0.876752	+	0.284874i	0.712608	−	0.701562i	\(-0.247514\pi\)
							0.164144	+	0.986436i	\(0.447514\pi\)
\(43\)	2.79832	+	2.79832i	0.426739	+	0.426739i	0.887516	−	0.460777i	\(-0.152429\pi\)
							−0.460777	+	0.887516i	\(0.652429\pi\)
\(47\)	−1.75513	−	0.277985i	−0.256011	−	0.0405482i	0.0271094	−	0.999632i	\(-0.491370\pi\)
							−0.283121	+	0.959084i	\(0.591370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bk.a.8.18	✓	480
3.2	odd	2	inner	525.2.bk.a.8.43	yes	480
25.22	odd	20	inner	525.2.bk.a.197.43	yes	480
75.47	even	20	inner	525.2.bk.a.197.18	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bk.a.8.18	✓	480	1.1	even	1	trivial
525.2.bk.a.8.43	yes	480	3.2	odd	2	inner
525.2.bk.a.197.18	yes	480	75.47	even	20	inner
525.2.bk.a.197.43	yes	480	25.22	odd	20	inner