Embedded newform 539.2.q.b.422.2

• Label: 539.2.q.b.422.2
• Level: $539$
• Weight: $2$
• Character: 539.422
• 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			422.2
Root			\(1.43468 + 0.304951i\) of defining polynomial
Character	\(\chi\)	\(=\)	539.422
Dual form			539.2.q.b.410.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.25347 - 1.00331i) q^{2} +(-1.58268 + 0.336408i) q^{3} +(2.73324 - 3.03557i) q^{4} +(-0.362372 - 3.44774i) q^{5} +(-3.22899 + 2.34600i) q^{6} +(1.58914 - 4.89086i) q^{8} +(-0.348943 + 0.155360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} - 4 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.25347	−	1.00331i	1.59345	−	0.709447i	0.597711	−	0.801712i	\(-0.296077\pi\)
							0.995735	+	0.0922645i	\(0.0294105\pi\)
\(3\)	−1.58268	+	0.336408i	−0.913758	+	0.194225i	−0.640717	−	0.767777i	\(-0.721363\pi\)
							−0.273041	+	0.962002i	\(0.588030\pi\)
\(5\)	−0.362372	−	3.44774i	−0.162058	−	1.54188i	−0.709303	−	0.704904i	\(-0.750990\pi\)
							0.547245	−	0.836972i	\(-0.315677\pi\)
\(7\)		0			0
							\(11\)	−2.82208	−	1.74237i	−0.850890	−	0.525344i
\(13\)	−0.528896	−	0.384266i	−0.146689	−	0.106576i								0.512020	−	0.858973i	\(-0.328897\pi\)
							−0.658709	+	0.752397i	\(0.728897\pi\)
\(17\)	1.03884	+	0.462522i	0.251956	+	0.112178i	0.528829	−	0.848729i	\(-0.322631\pi\)
							−0.276873	+	0.960907i	\(0.589298\pi\)
\(19\)	4.06565	+	4.51536i	0.932725	+	1.03590i	0.999273	+	0.0381124i	\(0.0121345\pi\)
							−0.0665489	+	0.997783i	\(0.521199\pi\)
\(23\)	3.33354	−	5.77386i	0.695091	−	1.20393i	−0.275059	−	0.961427i	\(-0.588697\pi\)
							0.970150	−	0.242506i	\(-0.0779694\pi\)
\(29\)	−1.41331	−	4.34973i	−0.262445	−	0.807724i	−0.992271	−	0.124090i	\(-0.960399\pi\)
							0.729826	−	0.683633i	\(-0.239601\pi\)
\(31\)	0.292294	−	2.78099i	0.0524975	−	0.499481i	−0.936406	−	0.350919i	\(-0.885869\pi\)
							0.988903	−	0.148561i	\(-0.0474642\pi\)
\(37\)	0.430149	+	0.0914309i	0.0707160	+	0.0150311i	0.243134	−	0.969993i	\(-0.421825\pi\)
							−0.172418	+	0.985024i	\(0.555158\pi\)
\(41\)	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	\(0.268860\pi\)
\(47\)	0.404626	+	0.449382i	0.0590207	+	0.0655491i	0.771932	−	0.635705i	\(-0.219291\pi\)
							−0.712911	+	0.701254i	\(0.752624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.b.422.2		16
7.2	even	3		539.2.f.d.246.1		8
7.3	odd	6		539.2.q.c.312.1		16
7.4	even	3	inner	539.2.q.b.312.1		16
7.5	odd	6		77.2.f.a.15.1	✓	8
7.6	odd	2		539.2.q.c.422.2		16
11.3	even	5	inner	539.2.q.b.520.1		16
21.5	even	6		693.2.m.g.631.2		8
77.3	odd	30		539.2.q.c.410.2		16
77.5	odd	30		847.2.a.l.1.4		4
77.16	even	15		5929.2.a.bi.1.4		4
77.19	even	30		847.2.f.q.729.2		8
77.25	even	15	inner	539.2.q.b.410.2		16
77.26	odd	30		847.2.f.p.372.2		8
77.40	even	30		847.2.f.s.372.1		8
77.47	odd	30		77.2.f.a.36.1	yes	8
77.54	even	6		847.2.f.q.323.2		8
77.58	even	15		539.2.f.d.344.1		8
77.61	even	30		847.2.a.k.1.1		4
77.68	even	30		847.2.f.s.148.1		8
77.69	odd	10		539.2.q.c.520.1		16
77.72	odd	30		5929.2.a.bb.1.1		4
77.75	odd	30		847.2.f.p.148.2		8
231.5	even	30		7623.2.a.ch.1.1		4
231.47	even	30		693.2.m.g.190.2		8
231.215	odd	30		7623.2.a.co.1.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.1	✓	8	7.5	odd	6
77.2.f.a.36.1	yes	8	77.47	odd	30
539.2.f.d.246.1		8	7.2	even	3
539.2.f.d.344.1		8	77.58	even	15
539.2.q.b.312.1		16	7.4	even	3	inner
539.2.q.b.410.2		16	77.25	even	15	inner
539.2.q.b.422.2		16	1.1	even	1	trivial
539.2.q.b.520.1		16	11.3	even	5	inner
539.2.q.c.312.1		16	7.3	odd	6
539.2.q.c.410.2		16	77.3	odd	30
539.2.q.c.422.2		16	7.6	odd	2
539.2.q.c.520.1		16	77.69	odd	10
693.2.m.g.190.2		8	231.47	even	30
693.2.m.g.631.2		8	21.5	even	6
847.2.a.k.1.1		4	77.61	even	30
847.2.a.l.1.4		4	77.5	odd	30
847.2.f.p.148.2		8	77.75	odd	30
847.2.f.p.372.2		8	77.26	odd	30
847.2.f.q.323.2		8	77.54	even	6
847.2.f.q.729.2		8	77.19	even	30
847.2.f.s.148.1		8	77.68	even	30
847.2.f.s.372.1		8	77.40	even	30
5929.2.a.bb.1.1		4	77.72	odd	30
5929.2.a.bi.1.4		4	77.16	even	15
7623.2.a.ch.1.1		4	231.5	even	30
7623.2.a.co.1.4		4	231.215	odd	30