Embedded newform 539.2.q.b.312.1

• Label: 539.2.q.b.312.1
• Level: $539$
• Weight: $2$
• Character: 539.312
• 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			312.1
Root			\(-0.981435 + 1.08999i\) of defining polynomial
Character	\(\chi\)	\(=\)	539.312
Dual form			539.2.q.b.520.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.257844 + 2.45322i) q^{2} +(1.08268 + 1.20243i) q^{3} +(-3.99550 - 0.849271i) q^{4} +(3.16702 + 1.41005i) q^{5} +(-3.22899 + 2.34600i) q^{6} +(1.58914 - 4.89086i) q^{8} +(0.0399263 - 0.379874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} - 4 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.257844	+	2.45322i	−0.182323	+	1.73469i	0.395448	+	0.918489i	\(0.370589\pi\)
							−0.577771	+	0.816199i	\(0.696077\pi\)
\(3\)	1.08268	+	1.20243i	0.625083	+	0.694225i	0.969639	−	0.244540i	\(-0.0786371\pi\)
							−0.344556	+	0.938766i	\(0.611970\pi\)
\(5\)	3.16702	+	1.41005i	1.41633	+	0.630592i	0.965116	−	0.261822i	\(-0.0843234\pi\)
							0.451217	+	0.892414i	\(0.350990\pi\)
\(7\)		0			0
							\(11\)	−0.0978940	+	3.31518i	−0.0295161	+	0.999564i
\(13\)	−0.528896	−	0.384266i	−0.146689	−	0.106576i								0.512020	−	0.858973i	\(-0.328897\pi\)
							−0.658709	+	0.752397i	\(0.728897\pi\)
\(17\)	−0.118865	−	1.13092i	−0.0288290	−	0.274289i	−0.999435	−	0.0336149i	\(-0.989298\pi\)
							0.970606	−	0.240675i	\(-0.0773686\pi\)
\(19\)	−5.94325	+	1.26328i	−1.36347	+	0.289815i	−0.830831	−	0.556525i	\(-0.812134\pi\)
							−0.532643	+	0.846340i	\(0.678801\pi\)
\(23\)	3.33354	+	5.77386i	0.695091	+	1.20393i	0.970150	+	0.242506i	\(0.0779694\pi\)
							−0.275059	+	0.961427i	\(0.588697\pi\)
\(29\)	−1.41331	−	4.34973i	−0.262445	−	0.807724i	−0.992271	−	0.124090i	\(-0.960399\pi\)
							0.729826	−	0.683633i	\(-0.239601\pi\)
\(31\)	−2.55456	+	1.13736i	−0.458812	+	0.204276i	−0.623109	−	0.782135i	\(-0.714131\pi\)
							0.164298	+	0.986411i	\(0.447464\pi\)
\(37\)	−0.294256	+	0.326804i	−0.0483753	+	0.0537263i	−0.766847	−	0.641830i	\(-0.778175\pi\)
							0.718471	+	0.695556i	\(0.244842\pi\)
\(41\)	1.82417	−	5.61423i	0.284888	−	0.876795i	−0.701544	−	0.712626i	\(-0.747506\pi\)
							0.986432	−	0.164169i	\(-0.0524943\pi\)
\(43\)	8.70820			1.32799			0.663994	−	0.747738i	\(-0.268860\pi\)
							0.663994	+	0.747738i	\(0.268860\pi\)
\(47\)	−0.591489	+	0.125725i	−0.0862776	+	0.0183389i	−0.250848	−	0.968026i	\(-0.580709\pi\)
							0.164571	+	0.986365i	\(0.447376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.b.312.1		16
7.2	even	3	inner	539.2.q.b.422.2		16
7.3	odd	6		77.2.f.a.15.1	✓	8
7.4	even	3		539.2.f.d.246.1		8
7.5	odd	6		539.2.q.c.422.2		16
7.6	odd	2		539.2.q.c.312.1		16
11.3	even	5	inner	539.2.q.b.410.2		16
21.17	even	6		693.2.m.g.631.2		8
77.3	odd	30		77.2.f.a.36.1	yes	8
77.10	even	6		847.2.f.q.323.2		8
77.17	even	30		847.2.a.k.1.1		4
77.24	even	30		847.2.f.s.148.1		8
77.25	even	15		539.2.f.d.344.1		8
77.31	odd	30		847.2.f.p.148.2		8
77.38	odd	30		847.2.a.l.1.4		4
77.39	odd	30		5929.2.a.bb.1.1		4
77.47	odd	30		539.2.q.c.520.1		16
77.52	even	30		847.2.f.q.729.2		8
77.58	even	15	inner	539.2.q.b.520.1		16
77.59	odd	30		847.2.f.p.372.2		8
77.60	even	15		5929.2.a.bi.1.4		4
77.69	odd	10		539.2.q.c.410.2		16
77.73	even	30		847.2.f.s.372.1		8
231.17	odd	30		7623.2.a.co.1.4		4
231.38	even	30		7623.2.a.ch.1.1		4
231.80	even	30		693.2.m.g.190.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.1	✓	8	7.3	odd	6
77.2.f.a.36.1	yes	8	77.3	odd	30
539.2.f.d.246.1		8	7.4	even	3
539.2.f.d.344.1		8	77.25	even	15
539.2.q.b.312.1		16	1.1	even	1	trivial
539.2.q.b.410.2		16	11.3	even	5	inner
539.2.q.b.422.2		16	7.2	even	3	inner
539.2.q.b.520.1		16	77.58	even	15	inner
539.2.q.c.312.1		16	7.6	odd	2
539.2.q.c.410.2		16	77.69	odd	10
539.2.q.c.422.2		16	7.5	odd	6
539.2.q.c.520.1		16	77.47	odd	30
693.2.m.g.190.2		8	231.80	even	30
693.2.m.g.631.2		8	21.17	even	6
847.2.a.k.1.1		4	77.17	even	30
847.2.a.l.1.4		4	77.38	odd	30
847.2.f.p.148.2		8	77.31	odd	30
847.2.f.p.372.2		8	77.59	odd	30
847.2.f.q.323.2		8	77.10	even	6
847.2.f.q.729.2		8	77.52	even	30
847.2.f.s.148.1		8	77.24	even	30
847.2.f.s.372.1		8	77.73	even	30
5929.2.a.bb.1.1		4	77.39	odd	30
5929.2.a.bi.1.4		4	77.60	even	15
7623.2.a.ch.1.1		4	231.38	even	30
7623.2.a.co.1.4		4	231.17	odd	30