Embedded newform 539.2.q.g.324.1

• Label: 539.2.q.g.324.1
• Level: $539$
• Weight: $2$
• Character: 539.324
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $32$
• 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:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{15})\)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			324.1
Character	\(\chi\)	\(=\)	539.324
Dual form			539.2.q.g.361.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49618 - 0.530579i) q^{2} +(0.0429192 - 0.408349i) q^{3} +(4.12229 + 1.83536i) q^{4} +(2.29443 + 2.54823i) q^{5} +(-0.323795 + 0.996539i) q^{6} +(-5.18703 - 3.76860i) q^{8} +(2.76954 + 0.588683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 3 q^{2} + 2 q^{3} + 11 q^{4} + 5 q^{5} + 6 q^{6} - 10 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49618	−	0.530579i	−1.76506	−	0.375176i	−0.792876	−	0.609383i	\(-0.791417\pi\)
							−0.972187	+	0.234207i	\(0.924751\pi\)
\(3\)	0.0429192	−	0.408349i	0.0247794	−	0.235760i	−0.975121	−	0.221672i	\(-0.928849\pi\)
							0.999901	−	0.0140886i	\(-0.00448469\pi\)
\(5\)	2.29443	+	2.54823i	1.02610	+	1.13960i	0.990116	+	0.140249i	\(0.0447903\pi\)
							0.0359855	+	0.999352i	\(0.488543\pi\)
\(7\)		0			0
							\(11\)	2.88705	−	1.63246i	0.870479	−	0.492206i
\(13\)	−0.672122	−	2.06858i	−0.186413	−	0.573720i								0.813557	−	0.581486i	\(-0.197528\pi\)
							−0.999970	+	0.00776501i	\(0.997528\pi\)
\(17\)	4.41844	−	0.939168i	1.07163	−	0.227782i	0.361871	−	0.932228i	\(-0.382138\pi\)
							0.709757	+	0.704446i	\(0.248805\pi\)
\(19\)	−2.21426	+	0.985853i	−0.507986	+	0.226170i	−0.644693	−	0.764441i	\(-0.723015\pi\)
							0.136707	+	0.990612i	\(0.456348\pi\)
\(23\)	−0.324201	+	0.561533i	−0.0676006	+	0.117088i	−0.897845	−	0.440312i	\(-0.854868\pi\)
							0.830244	+	0.557400i	\(0.188201\pi\)
\(29\)	1.01366	−	0.736466i	0.188232	−	0.136758i	−0.489678	−	0.871903i	\(-0.662886\pi\)
							0.677910	+	0.735145i	\(0.262886\pi\)
\(31\)	−5.37945	+	5.97449i	−0.966178	+	1.07305i	0.0311146	+	0.999516i	\(0.490094\pi\)
							−0.997293	+	0.0735337i	\(0.976572\pi\)
\(37\)	−0.524032	−	4.98583i	−0.0861503	−	0.819665i	−0.949227	−	0.314592i	\(-0.898132\pi\)
							0.863077	−	0.505073i	\(-0.168534\pi\)
\(41\)	2.12100	+	1.54100i	0.331245	+	0.240664i	0.740959	−	0.671551i	\(-0.234371\pi\)
							−0.409714	+	0.912214i	\(0.634371\pi\)
\(43\)	−1.46138			−0.222858			−0.111429	−	0.993772i	\(-0.535543\pi\)
							−0.111429	+	0.993772i	\(0.535543\pi\)
\(47\)	−4.60462	+	2.05011i	−0.671653	+	0.299039i	−0.714084	−	0.700060i	\(-0.753157\pi\)
							0.0424305	+	0.999099i	\(0.486490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.g.324.1		32
7.2	even	3		77.2.f.b.71.4	yes	16
7.3	odd	6		539.2.q.f.214.4		32
7.4	even	3	inner	539.2.q.g.214.4		32
7.5	odd	6		539.2.f.e.148.4		16
7.6	odd	2		539.2.q.f.324.1		32
11.9	even	5	inner	539.2.q.g.471.4		32
21.2	odd	6		693.2.m.i.379.1		16
77.2	odd	30		847.2.f.x.372.1		16
77.9	even	15		77.2.f.b.64.4	✓	16
77.16	even	15		847.2.f.w.323.1		16
77.19	even	30		5929.2.a.bs.1.1		8
77.20	odd	10		539.2.q.f.471.4		32
77.30	odd	30		847.2.a.o.1.1		8
77.31	odd	30		539.2.q.f.361.1		32
77.37	even	15		847.2.f.w.729.1		16
77.47	odd	30		5929.2.a.bt.1.8		8
77.51	odd	30		847.2.f.v.729.4		16
77.53	even	15	inner	539.2.q.g.361.1		32
77.58	even	15		847.2.a.p.1.8		8
77.65	odd	6		847.2.f.x.148.1		16
77.72	odd	30		847.2.f.v.323.4		16
77.75	odd	30		539.2.f.e.295.4		16
231.86	odd	30		693.2.m.i.64.1		16
231.107	even	30		7623.2.a.cw.1.8		8
231.212	odd	30		7623.2.a.ct.1.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.4	✓	16	77.9	even	15
77.2.f.b.71.4	yes	16	7.2	even	3
539.2.f.e.148.4		16	7.5	odd	6
539.2.f.e.295.4		16	77.75	odd	30
539.2.q.f.214.4		32	7.3	odd	6
539.2.q.f.324.1		32	7.6	odd	2
539.2.q.f.361.1		32	77.31	odd	30
539.2.q.f.471.4		32	77.20	odd	10
539.2.q.g.214.4		32	7.4	even	3	inner
539.2.q.g.324.1		32	1.1	even	1	trivial
539.2.q.g.361.1		32	77.53	even	15	inner
539.2.q.g.471.4		32	11.9	even	5	inner
693.2.m.i.64.1		16	231.86	odd	30
693.2.m.i.379.1		16	21.2	odd	6
847.2.a.o.1.1		8	77.30	odd	30
847.2.a.p.1.8		8	77.58	even	15
847.2.f.v.323.4		16	77.72	odd	30
847.2.f.v.729.4		16	77.51	odd	30
847.2.f.w.323.1		16	77.16	even	15
847.2.f.w.729.1		16	77.37	even	15
847.2.f.x.148.1		16	77.65	odd	6
847.2.f.x.372.1		16	77.2	odd	30
5929.2.a.bs.1.1		8	77.19	even	30
5929.2.a.bt.1.8		8	77.47	odd	30
7623.2.a.ct.1.1		8	231.212	odd	30
7623.2.a.cw.1.8		8	231.107	even	30