Embedded newform 539.2.q.c.520.2

• Label: 539.2.q.c.520.2
• Level: $539$
• Weight: $2$
• Character: 539.520
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.q (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{15})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - x^15 - 5*x^14 + 16*x^13 + 4*x^12 - 29*x^11 + 10*x^10 - 156*x^9 + 251*x^8 + 240*x^7 + 390*x^6 - 1375*x^5 - 300*x^4 + 500*x^3 + 375*x^2 + 625*x + 625
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			520.2
Root			\(1.65057 + 1.83314i\) of defining polynomial
Character	\(\chi\)	\(=\)	539.520
Dual form			539.2.q.c.312.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.153315 + 1.45870i) q^{2} +(-1.08268 + 1.20243i) q^{3} +(-0.147996 + 0.0314575i) q^{4} +(0.426381 - 0.189837i) q^{5} +(-1.91998 - 1.39494i) q^{6} +(0.837913 + 2.57883i) q^{8} +(0.0399263 + 0.379874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 4 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.153315	+	1.45870i	0.108410	+	1.03145i	0.904558	+	0.426352i	\(0.140201\pi\)
							−0.796147	+	0.605103i	\(0.793132\pi\)
\(3\)	−1.08268	+	1.20243i	−0.625083	+	0.694225i	−0.969639	−	0.244540i	\(-0.921363\pi\)
							0.344556	+	0.938766i	\(0.388030\pi\)
\(5\)	0.426381	−	0.189837i	0.190683	−	0.0848977i	−0.309174	−	0.951006i	\(-0.600053\pi\)
							0.499857	+	0.866108i	\(0.333386\pi\)
\(7\)		0			0
							\(11\)	3.24139	+	0.702401i	0.977317	+	0.211782i
\(13\)	1.28012	−	0.930062i	0.355042	−	0.257953i								−0.395939	−	0.918277i	\(-0.629581\pi\)
							0.750981	+	0.660324i	\(0.229581\pi\)
\(17\)	−0.546404	+	5.19869i	−0.132522	+	1.26087i	0.702912	+	0.711277i	\(0.251883\pi\)
							−0.835434	+	0.549590i	\(0.814784\pi\)
\(19\)	−4.12966	−	0.877786i	−0.947409	−	0.201378i	−0.291801	−	0.956479i	\(-0.594254\pi\)
							−0.655608	+	0.755101i	\(0.727588\pi\)
\(23\)	0.902527	−	1.56322i	0.188190	−	0.325954i	−0.756457	−	0.654044i	\(-0.773071\pi\)
							0.944647	+	0.328089i	\(0.106405\pi\)
\(29\)	0.840363	−	2.58637i	0.156051	−	0.480277i	−0.842215	−	0.539143i	\(-0.818748\pi\)
							0.998266	+	0.0588657i	\(0.0187484\pi\)
\(31\)	1.18200	+	0.526260i	0.212293	+	0.0945192i	0.510128	−	0.860098i	\(-0.329598\pi\)
							−0.297835	+	0.954617i	\(0.596265\pi\)
\(37\)	−1.29959	−	1.44334i	−0.213651	−	0.237284i	0.626788	−	0.779190i	\(-0.284369\pi\)
							−0.840439	+	0.541906i	\(0.817703\pi\)
\(41\)	−0.321724	−	0.990166i	−0.0502449	−	0.154638i	0.922786	−	0.385313i	\(-0.125907\pi\)
							−0.973031	+	0.230675i	\(0.925907\pi\)
\(43\)	8.70820			1.32799			0.663994	−	0.747738i	\(-0.268860\pi\)
							0.663994	+	0.747738i	\(0.268860\pi\)
\(47\)	6.25554	+	1.32966i	0.912465	+	0.193950i	0.640143	−	0.768256i	\(-0.278875\pi\)
							0.272322	+	0.962206i	\(0.412208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.c.520.2		16
7.2	even	3		77.2.f.a.36.2	yes	8
7.3	odd	6		539.2.q.b.410.1		16
7.4	even	3	inner	539.2.q.c.410.1		16
7.5	odd	6		539.2.f.d.344.2		8
7.6	odd	2		539.2.q.b.520.2		16
11.4	even	5	inner	539.2.q.c.422.1		16
21.2	odd	6		693.2.m.g.190.1		8
77.2	odd	30		847.2.a.k.1.4		4
77.4	even	15	inner	539.2.q.c.312.2		16
77.9	even	15		847.2.a.l.1.1		4
77.16	even	15		847.2.f.p.372.1		8
77.26	odd	30		539.2.f.d.246.2		8
77.30	odd	30		847.2.f.s.148.2		8
77.37	even	15		77.2.f.a.15.2	✓	8
77.48	odd	10		539.2.q.b.422.1		16
77.51	odd	30		847.2.f.q.323.1		8
77.58	even	15		847.2.f.p.148.1		8
77.59	odd	30		539.2.q.b.312.2		16
77.65	odd	6		847.2.f.q.729.1		8
77.68	even	30		5929.2.a.bb.1.4		4
77.72	odd	30		847.2.f.s.372.2		8
77.75	odd	30		5929.2.a.bi.1.1		4
231.2	even	30		7623.2.a.co.1.1		4
231.86	odd	30		7623.2.a.ch.1.4		4
231.191	odd	30		693.2.m.g.631.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.2	✓	8	77.37	even	15
77.2.f.a.36.2	yes	8	7.2	even	3
539.2.f.d.246.2		8	77.26	odd	30
539.2.f.d.344.2		8	7.5	odd	6
539.2.q.b.312.2		16	77.59	odd	30
539.2.q.b.410.1		16	7.3	odd	6
539.2.q.b.422.1		16	77.48	odd	10
539.2.q.b.520.2		16	7.6	odd	2
539.2.q.c.312.2		16	77.4	even	15	inner
539.2.q.c.410.1		16	7.4	even	3	inner
539.2.q.c.422.1		16	11.4	even	5	inner
539.2.q.c.520.2		16	1.1	even	1	trivial
693.2.m.g.190.1		8	21.2	odd	6
693.2.m.g.631.1		8	231.191	odd	30
847.2.a.k.1.4		4	77.2	odd	30
847.2.a.l.1.1		4	77.9	even	15
847.2.f.p.148.1		8	77.58	even	15
847.2.f.p.372.1		8	77.16	even	15
847.2.f.q.323.1		8	77.51	odd	30
847.2.f.q.729.1		8	77.65	odd	6
847.2.f.s.148.2		8	77.30	odd	30
847.2.f.s.372.2		8	77.72	odd	30
5929.2.a.bb.1.4		4	77.68	even	30
5929.2.a.bi.1.1		4	77.75	odd	30
7623.2.a.ch.1.4		4	231.86	odd	30
7623.2.a.co.1.1		4	231.2	even	30