Embedded newform 539.2.q.b.471.1

• Label: 539.2.q.b.471.1
• Level: $539$
• Weight: $2$
• Character: 539.471
• 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			471.1
Root			\(1.62381 - 0.722968i\) of defining polynomial
Character	\(\chi\)	\(=\)	539.471
Dual form			539.2.q.b.214.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.520239 - 0.577783i) q^{2} +(-0.564602 - 0.251377i) q^{3} +(0.145871 - 1.38787i) q^{4} +(-0.217653 - 0.0462636i) q^{5} +(0.148486 + 0.456994i) q^{6} +(-2.13577 + 1.55173i) q^{8} +(-1.75181 - 1.94558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} - 4 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.520239	−	0.577783i	−0.367864	−	0.408555i	0.530585	−	0.847632i	\(-0.321972\pi\)
							−0.898449	+	0.439077i	\(0.855306\pi\)
\(3\)	−0.564602	−	0.251377i	−0.325973	−	0.145133i	0.237224	−	0.971455i	\(-0.423762\pi\)
							−0.563198	+	0.826322i	\(0.690429\pi\)
\(5\)	−0.217653	−	0.0462636i	−0.0973375	−	0.0206897i	0.158985	−	0.987281i	\(-0.449178\pi\)
							−0.256323	+	0.966591i	\(0.582511\pi\)
\(7\)		0			0
							\(11\)	−3.14501	+	1.05306i	−0.948255	+	0.317508i
\(13\)	−2.01774	+	6.20997i	−0.559620	+	1.72233i								0.123798	+	0.992307i	\(0.460493\pi\)
							−0.683418	+	0.730027i	\(0.739507\pi\)
\(17\)	2.90042	−	3.22125i	0.703456	−	0.781267i	−0.280466	−	0.959864i	\(-0.590489\pi\)
							0.983922	+	0.178597i	\(0.0571558\pi\)
\(19\)	0.304701	+	2.89904i	0.0699032	+	0.665085i	0.972229	+	0.234031i	\(0.0751916\pi\)
							−0.902326	+	0.431054i	\(0.858142\pi\)
\(23\)	1.94898	−	3.37573i	0.406390	−	0.703888i	−0.588092	−	0.808794i	\(-0.700121\pi\)
							0.994482	+	0.104906i	\(0.0334540\pi\)
\(29\)	3.05322	+	2.21829i	0.566969	+	0.411927i	0.834003	−	0.551760i	\(-0.186044\pi\)
							−0.267034	+	0.963687i	\(0.586044\pi\)
\(31\)	−6.73903	+	1.43242i	−1.21037	+	0.257271i	−0.768523	−	0.639822i	\(-0.779008\pi\)
							−0.441842	+	0.897093i	\(0.645675\pi\)
\(37\)	−5.16474	+	2.29949i	−0.849078	+	0.378034i	−0.784688	−	0.619891i	\(-0.787177\pi\)
							−0.0643903	+	0.997925i	\(0.520510\pi\)
\(41\)	1.08255	−	0.786521i	0.169066	−	0.122834i	−0.500035	−	0.866005i	\(-0.666679\pi\)
							0.669101	+	0.743171i	\(0.266679\pi\)
\(43\)	−4.70820			−0.717994			−0.358997	−	0.933339i	\(-0.616881\pi\)
							−0.358997	+	0.933339i	\(0.616881\pi\)
\(47\)	0.631931	+	6.01242i	0.0921765	+	0.877001i	0.938720	+	0.344681i	\(0.112013\pi\)
							−0.846543	+	0.532320i	\(0.821320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.b.471.1		16
7.2	even	3		539.2.f.d.295.1		8
7.3	odd	6		539.2.q.c.361.2		16
7.4	even	3	inner	539.2.q.b.361.2		16
7.5	odd	6		77.2.f.a.64.1	✓	8
7.6	odd	2		539.2.q.c.471.1		16
11.5	even	5	inner	539.2.q.b.324.2		16
21.5	even	6		693.2.m.g.64.2		8
77.5	odd	30		77.2.f.a.71.1	yes	8
77.16	even	15		539.2.f.d.148.1		8
77.19	even	30		847.2.f.s.323.1		8
77.26	odd	30		847.2.a.l.1.2		4
77.27	odd	10		539.2.q.c.324.2		16
77.37	even	15		5929.2.a.bi.1.2		4
77.38	odd	30		539.2.q.c.214.1		16
77.40	even	30		847.2.a.k.1.3		4
77.47	odd	30		847.2.f.p.323.2		8
77.51	odd	30		5929.2.a.bb.1.3		4
77.54	even	6		847.2.f.q.372.2		8
77.60	even	15	inner	539.2.q.b.214.1		16
77.61	even	30		847.2.f.q.148.2		8
77.68	even	30		847.2.f.s.729.1		8
77.75	odd	30		847.2.f.p.729.2		8
231.5	even	30		693.2.m.g.379.2		8
231.26	even	30		7623.2.a.ch.1.3		4
231.194	odd	30		7623.2.a.co.1.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.64.1	✓	8	7.5	odd	6
77.2.f.a.71.1	yes	8	77.5	odd	30
539.2.f.d.148.1		8	77.16	even	15
539.2.f.d.295.1		8	7.2	even	3
539.2.q.b.214.1		16	77.60	even	15	inner
539.2.q.b.324.2		16	11.5	even	5	inner
539.2.q.b.361.2		16	7.4	even	3	inner
539.2.q.b.471.1		16	1.1	even	1	trivial
539.2.q.c.214.1		16	77.38	odd	30
539.2.q.c.324.2		16	77.27	odd	10
539.2.q.c.361.2		16	7.3	odd	6
539.2.q.c.471.1		16	7.6	odd	2
693.2.m.g.64.2		8	21.5	even	6
693.2.m.g.379.2		8	231.5	even	30
847.2.a.k.1.3		4	77.40	even	30
847.2.a.l.1.2		4	77.26	odd	30
847.2.f.p.323.2		8	77.47	odd	30
847.2.f.p.729.2		8	77.75	odd	30
847.2.f.q.148.2		8	77.61	even	30
847.2.f.q.372.2		8	77.54	even	6
847.2.f.s.323.1		8	77.19	even	30
847.2.f.s.729.1		8	77.68	even	30
5929.2.a.bb.1.3		4	77.51	odd	30
5929.2.a.bi.1.2		4	77.37	even	15
7623.2.a.ch.1.3		4	231.26	even	30
7623.2.a.co.1.2		4	231.194	odd	30