Embedded newform 539.2.q.c.324.1

• Label: 539.2.q.c.324.1
• Level: $539$
• Weight: $2$
• Character: 539.324
• 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			324.1
Root			\(0.0812692 + 0.773225i\) of defining polynomial
Character	\(\chi\)	\(=\)	539.324
Dual form			539.2.q.c.361.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73864 - 0.369560i) q^{2} +(-0.0646021 + 0.614648i) q^{3} +(1.05921 + 0.471591i) q^{4} +(-1.85850 - 2.06407i) q^{5} +(0.339469 - 1.04478i) q^{6} +(1.20872 + 0.878189i) q^{8} +(2.56082 + 0.544320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 4 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73864	−	0.369560i	−1.22941	−	0.261318i	−0.452976	−	0.891523i	\(-0.649638\pi\)
							−0.776429	+	0.630204i	\(0.782971\pi\)
\(3\)	−0.0646021	+	0.614648i	−0.0372981	+	0.354867i	0.959917	+	0.280285i	\(0.0904290\pi\)
							−0.997215	+	0.0745823i	\(0.976238\pi\)
\(5\)	−1.85850	−	2.06407i	−0.831146	−	0.923082i	0.166874	−	0.985978i	\(-0.446633\pi\)
							−0.998020	+	0.0628968i	\(0.979966\pi\)
\(7\)		0			0
							\(11\)	−3.26882	+	0.561095i	−0.985586	+	0.169177i
\(13\)	−1.32676	−	4.08334i	−0.367976	−	1.13251i								−0.948096	−	0.317983i	\(-0.896994\pi\)
							0.580120	−	0.814531i	\(-0.303006\pi\)
\(17\)	2.69534	−	0.572912i	0.653716	−	0.138952i	0.130898	−	0.991396i	\(-0.458214\pi\)
							0.522818	+	0.852444i	\(0.324881\pi\)
\(19\)	1.77145	−	0.788701i	0.406399	−	0.180940i	−0.193348	−	0.981130i	\(-0.561935\pi\)
							0.599747	+	0.800190i	\(0.295268\pi\)
\(23\)	−2.18505	+	3.78461i	−0.455614	+	0.789146i	−0.998723	−	0.0505157i	\(-0.983914\pi\)
							0.543109	+	0.839662i	\(0.317247\pi\)
\(29\)	−6.98027	+	5.07146i	−1.29620	+	0.941747i	−0.999911	−	0.0133499i	\(-0.995750\pi\)
							−0.296293	+	0.955097i	\(0.595750\pi\)
\(31\)	−0.134219	+	0.149066i	−0.0241065	+	0.0267730i	−0.755078	−	0.655636i	\(-0.772401\pi\)
							0.730971	+	0.682409i	\(0.239067\pi\)
\(37\)	−0.108237	−	1.02981i	−0.0177941	−	0.169299i	0.982016	−	0.188799i	\(-0.0604594\pi\)
							−0.999810	+	0.0194995i	\(0.993793\pi\)
\(41\)	−7.77155	−	5.64636i	−1.21371	−	0.881813i	−0.218149	−	0.975915i	\(-0.570002\pi\)
							−0.995563	+	0.0941021i	\(0.970002\pi\)
\(43\)	−4.70820			−0.717994			−0.358997	−	0.933339i	\(-0.616881\pi\)
							−0.358997	+	0.933339i	\(0.616881\pi\)
\(47\)	−11.9177	+	5.30610i	−1.73837	+	0.773974i	−0.743984	+	0.668198i	\(0.767066\pi\)
							−0.994390	+	0.105776i	\(0.966267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.c.324.1		16
7.2	even	3		77.2.f.a.71.2	yes	8
7.3	odd	6		539.2.q.b.214.2		16
7.4	even	3	inner	539.2.q.c.214.2		16
7.5	odd	6		539.2.f.d.148.2		8
7.6	odd	2		539.2.q.b.324.1		16
11.9	even	5	inner	539.2.q.c.471.2		16
21.2	odd	6		693.2.m.g.379.1		8
77.2	odd	30		847.2.f.q.372.1		8
77.9	even	15		77.2.f.a.64.2	✓	8
77.16	even	15		847.2.f.p.323.1		8
77.19	even	30		5929.2.a.bb.1.2		4
77.20	odd	10		539.2.q.b.471.2		16
77.30	odd	30		847.2.a.k.1.2		4
77.31	odd	30		539.2.q.b.361.1		16
77.37	even	15		847.2.f.p.729.1		8
77.47	odd	30		5929.2.a.bi.1.3		4
77.51	odd	30		847.2.f.s.729.2		8
77.53	even	15	inner	539.2.q.c.361.1		16
77.58	even	15		847.2.a.l.1.3		4
77.65	odd	6		847.2.f.q.148.1		8
77.72	odd	30		847.2.f.s.323.2		8
77.75	odd	30		539.2.f.d.295.2		8
231.86	odd	30		693.2.m.g.64.1		8
231.107	even	30		7623.2.a.co.1.3		4
231.212	odd	30		7623.2.a.ch.1.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.64.2	✓	8	77.9	even	15
77.2.f.a.71.2	yes	8	7.2	even	3
539.2.f.d.148.2		8	7.5	odd	6
539.2.f.d.295.2		8	77.75	odd	30
539.2.q.b.214.2		16	7.3	odd	6
539.2.q.b.324.1		16	7.6	odd	2
539.2.q.b.361.1		16	77.31	odd	30
539.2.q.b.471.2		16	77.20	odd	10
539.2.q.c.214.2		16	7.4	even	3	inner
539.2.q.c.324.1		16	1.1	even	1	trivial
539.2.q.c.361.1		16	77.53	even	15	inner
539.2.q.c.471.2		16	11.9	even	5	inner
693.2.m.g.64.1		8	231.86	odd	30
693.2.m.g.379.1		8	21.2	odd	6
847.2.a.k.1.2		4	77.30	odd	30
847.2.a.l.1.3		4	77.58	even	15
847.2.f.p.323.1		8	77.16	even	15
847.2.f.p.729.1		8	77.37	even	15
847.2.f.q.148.1		8	77.65	odd	6
847.2.f.q.372.1		8	77.2	odd	30
847.2.f.s.323.2		8	77.72	odd	30
847.2.f.s.729.2		8	77.51	odd	30
5929.2.a.bb.1.2		4	77.19	even	30
5929.2.a.bi.1.3		4	77.47	odd	30
7623.2.a.ch.1.2		4	231.212	odd	30
7623.2.a.co.1.3		4	231.107	even	30