Embedded newform 539.2.q.g.361.1

• Label: 539.2.q.g.361.1
• Level: $539$
• Weight: $2$
• Character: 539.361
• 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			361.1
Character	\(\chi\)	\(=\)	539.361
Dual form			539.2.q.g.324.1

$q$-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 \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.g.361.1		32
7.2	even	3	inner	539.2.q.g.471.4		32
7.3	odd	6		539.2.f.e.295.4		16
7.4	even	3		77.2.f.b.64.4	✓	16
7.5	odd	6		539.2.q.f.471.4		32
7.6	odd	2		539.2.q.f.361.1		32
11.5	even	5	inner	539.2.q.g.214.4		32
21.11	odd	6		693.2.m.i.64.1		16
77.4	even	15		847.2.a.p.1.8		8
77.5	odd	30		539.2.q.f.324.1		32
77.16	even	15	inner	539.2.q.g.324.1		32
77.18	odd	30		847.2.a.o.1.1		8
77.25	even	15		847.2.f.w.323.1		16
77.27	odd	10		539.2.q.f.214.4		32
77.32	odd	6		847.2.f.x.372.1		16
77.38	odd	30		539.2.f.e.148.4		16
77.39	odd	30		847.2.f.x.148.1		16
77.46	odd	30		847.2.f.v.729.4		16
77.53	even	15		847.2.f.w.729.1		16
77.59	odd	30		5929.2.a.bt.1.8		8
77.60	even	15		77.2.f.b.71.4	yes	16
77.73	even	30		5929.2.a.bs.1.1		8
77.74	odd	30		847.2.f.v.323.4		16
231.95	even	30		7623.2.a.cw.1.8		8
231.137	odd	30		693.2.m.i.379.1		16
231.158	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	7.4	even	3
77.2.f.b.71.4	yes	16	77.60	even	15
539.2.f.e.148.4		16	77.38	odd	30
539.2.f.e.295.4		16	7.3	odd	6
539.2.q.f.214.4		32	77.27	odd	10
539.2.q.f.324.1		32	77.5	odd	30
539.2.q.f.361.1		32	7.6	odd	2
539.2.q.f.471.4		32	7.5	odd	6
539.2.q.g.214.4		32	11.5	even	5	inner
539.2.q.g.324.1		32	77.16	even	15	inner
539.2.q.g.361.1		32	1.1	even	1	trivial
539.2.q.g.471.4		32	7.2	even	3	inner
693.2.m.i.64.1		16	21.11	odd	6
693.2.m.i.379.1		16	231.137	odd	30
847.2.a.o.1.1		8	77.18	odd	30
847.2.a.p.1.8		8	77.4	even	15
847.2.f.v.323.4		16	77.74	odd	30
847.2.f.v.729.4		16	77.46	odd	30
847.2.f.w.323.1		16	77.25	even	15
847.2.f.w.729.1		16	77.53	even	15
847.2.f.x.148.1		16	77.39	odd	30
847.2.f.x.372.1		16	77.32	odd	6
5929.2.a.bs.1.1		8	77.73	even	30
5929.2.a.bt.1.8		8	77.59	odd	30
7623.2.a.ct.1.1		8	231.158	odd	30
7623.2.a.cw.1.8		8	231.95	even	30