Embedded newform 525.2.bp.a.59.9

• Label: 525.2.bp.a.59.9
• Level: $525$
• Weight: $2$
• Character: 525.59
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $608$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bp (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(608\)
Relative dimension:	\(76\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			59.9
Character	\(\chi\)	\(=\)	525.59
Dual form			525.2.bp.a.89.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235548 + 2.24109i) q^{2} +(-1.62356 + 0.603364i) q^{3} +(-3.01072 - 0.639948i) q^{4} +(-0.938365 - 2.02965i) q^{5} +(-0.969767 - 3.78067i) q^{6} +(-0.601362 + 2.57650i) q^{7} +(0.750653 - 2.31027i) q^{8} +(2.27190 - 1.95920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 608 q - 15 q^{3} + 66 q^{4} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.235548	+	2.24109i	−0.166558	+	1.58469i	0.517771	+	0.855519i	\(0.326762\pi\)
							−0.684329	+	0.729173i	\(0.739905\pi\)
\(3\)	−1.62356	+	0.603364i	−0.937364	+	0.348352i
							\(5\)	−0.938365	−	2.02965i	−0.419650	−	0.907686i
\(7\)	−0.601362	+	2.57650i	−0.227293	+	0.973826i
							\(11\)	−2.11073	−	4.74079i	−0.636410	−	1.42940i	−0.887212	−	0.461362i	\(-0.847361\pi\)
0.250801	−	0.968039i	\(-0.419306\pi\)
\(13\)	1.48919	−	1.08196i	0.413026	−	0.300081i	−0.361799	−	0.932256i	\(-0.617837\pi\)
							0.774826	+	0.632175i	\(0.217837\pi\)
\(17\)	5.46895	−	4.92427i	1.32642	−	1.19431i	0.361312	−	0.932445i	\(-0.382329\pi\)
							0.965104	−	0.261866i	\(-0.0843377\pi\)
\(19\)	0.756004	+	3.55672i	0.173439	+	0.815967i	0.975720	+	0.219021i	\(0.0702863\pi\)
							−0.802281	+	0.596947i	\(0.796380\pi\)
\(23\)	−0.276239	+	2.62824i	−0.0575999	+	0.548026i	0.927229	+	0.374496i	\(0.122184\pi\)
							−0.984829	+	0.173530i	\(0.944483\pi\)
\(29\)	−1.32173	+	0.429455i	−0.245439	+	0.0797479i	−0.429153	−	0.903232i	\(-0.641188\pi\)
							0.183714	+	0.982980i	\(0.441188\pi\)
\(31\)	1.21232	−	1.09158i	0.217740	−	0.196054i	−0.553041	−	0.833154i	\(-0.686533\pi\)
							0.770781	+	0.637100i	\(0.219866\pi\)
\(37\)	0.422919	−	0.949892i	0.0695275	−	0.156161i	−0.875439	−	0.483329i	\(-0.839428\pi\)
							0.944966	+	0.327168i	\(0.106094\pi\)
\(41\)	10.1980	−	7.40928i	1.59266	−	1.15713i	0.692641	−	0.721283i	\(-0.256447\pi\)
							0.900018	−	0.435852i	\(-0.143553\pi\)
\(43\)		−	6.99065i		−	1.06606i	−0.846095	−	0.533032i	\(-0.821053\pi\)
							0.846095	−	0.533032i	\(-0.178947\pi\)
\(47\)	4.83287	+	4.35153i	0.704946	+	0.634736i	0.941551	−	0.336869i	\(-0.109368\pi\)
							−0.236605	+	0.971606i	\(0.576035\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bp.a.59.9	✓	608
3.2	odd	2	inner	525.2.bp.a.59.68	yes	608
7.5	odd	6	inner	525.2.bp.a.509.68	yes	608
21.5	even	6	inner	525.2.bp.a.509.9	yes	608
25.14	even	10	inner	525.2.bp.a.164.9	yes	608
75.14	odd	10	inner	525.2.bp.a.164.68	yes	608
175.89	odd	30	inner	525.2.bp.a.89.68	yes	608
525.89	even	30	inner	525.2.bp.a.89.9	yes	608

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bp.a.59.9	✓	608	1.1	even	1	trivial
525.2.bp.a.59.68	yes	608	3.2	odd	2	inner
525.2.bp.a.89.9	yes	608	525.89	even	30	inner
525.2.bp.a.89.68	yes	608	175.89	odd	30	inner
525.2.bp.a.164.9	yes	608	25.14	even	10	inner
525.2.bp.a.164.68	yes	608	75.14	odd	10	inner
525.2.bp.a.509.9	yes	608	21.5	even	6	inner
525.2.bp.a.509.68	yes	608	7.5	odd	6	inner