Embedded newform 726.2.e.o.565.1

• Label: 726.2.e.o.565.1
• Level: $726$
• Weight: $2$
• Character: 726.565
• Analytic conductor: $5.797$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	726.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.79713918674\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			565.1
Root			\(0.809017 - 0.587785i\) of defining polynomial
Character	\(\chi\)	\(=\)	726.565
Dual form			726.2.e.o.487.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{2} +(-0.309017 + 0.951057i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-1.61803 + 1.17557i) q^{5} +(-0.809017 + 0.587785i) q^{6} +(1.23607 + 3.80423i) q^{7} +(-0.309017 + 0.951057i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - 4 q^{7} + q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\)	\(485\)	\(607\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.809017	+	0.587785i	0.572061	+	0.415627i
							\(3\)	−0.309017	+	0.951057i	−0.178411	+	0.549093i
\(5\)	−1.61803	+	1.17557i	−0.723607	+	0.525731i								−0.887535	−	0.460741i	\(-0.847584\pi\)
							0.163928	+	0.986472i	\(0.447584\pi\)
\(7\)	1.23607	+	3.80423i	0.467190	+	1.43786i	0.856208	+	0.516632i	\(0.172814\pi\)
							−0.389018	+	0.921230i	\(0.627186\pi\)
\(11\)		0			0
							\(13\)	−4.85410	−	3.52671i	−1.34629	−	0.978134i	−0.999187	−	0.0403050i	\(-0.987167\pi\)
−0.347098	−	0.937829i	\(-0.612833\pi\)
\(17\)	1.61803	−	1.17557i	0.392431	−	0.285118i	−0.374020	−	0.927421i	\(-0.622021\pi\)
							0.766451	+	0.642303i	\(0.222021\pi\)
\(19\)	−1.23607	+	3.80423i	−0.283573	+	0.872749i	0.703249	+	0.710943i	\(0.251732\pi\)
							−0.986823	+	0.161806i	\(0.948268\pi\)
\(23\)	4.00000			0.834058			0.417029	−	0.908893i	\(-0.363071\pi\)
							0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−1.85410	−	5.70634i	−0.344298	−	1.05964i	−0.961958	−	0.273196i	\(-0.911919\pi\)
							0.617660	−	0.786445i	\(-0.288081\pi\)
\(31\)		0			0		0.587785	−	0.809017i	\(-0.300000\pi\)
							−0.587785	+	0.809017i	\(0.700000\pi\)
\(37\)	1.85410	+	5.70634i	0.304812	+	0.938116i	0.979747	+	0.200239i	\(0.0641718\pi\)
							−0.674935	+	0.737878i	\(0.735828\pi\)
\(41\)	1.85410	−	5.70634i	0.289562	−	0.891180i	−0.695432	−	0.718592i	\(-0.744787\pi\)
							0.984994	−	0.172588i	\(-0.0552131\pi\)
\(43\)	−4.00000			−0.609994			−0.304997	−	0.952353i	\(-0.598656\pi\)
							−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−3.70820	+	11.4127i	−0.540897	+	1.66471i	0.189653	+	0.981851i	\(0.439264\pi\)
							−0.730550	+	0.682859i	\(0.760736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	726.2.e.o.565.1		4
11.2	odd	10		726.2.e.g.511.1		4
11.3	even	5	inner	726.2.e.o.487.1		4
11.4	even	5	inner	726.2.e.o.493.1		4
11.5	even	5		726.2.a.c.1.1		1
11.6	odd	10		66.2.a.b.1.1	✓	1
11.7	odd	10		726.2.e.g.493.1		4
11.8	odd	10		726.2.e.g.487.1		4
11.9	even	5	inner	726.2.e.o.511.1		4
11.10	odd	2		726.2.e.g.565.1		4
33.5	odd	10		2178.2.a.g.1.1		1
33.17	even	10		198.2.a.a.1.1		1
44.27	odd	10		5808.2.a.bc.1.1		1
44.39	even	10		528.2.a.j.1.1		1
55.17	even	20		1650.2.c.e.199.2		2
55.28	even	20		1650.2.c.e.199.1		2
55.39	odd	10		1650.2.a.k.1.1		1
77.6	even	10		3234.2.a.t.1.1		1
88.61	odd	10		2112.2.a.r.1.1		1
88.83	even	10		2112.2.a.e.1.1		1
99.50	even	30		1782.2.e.v.1189.1		2
99.61	odd	30		1782.2.e.e.595.1		2
99.83	even	30		1782.2.e.v.595.1		2
99.94	odd	30		1782.2.e.e.1189.1		2
132.83	odd	10		1584.2.a.f.1.1		1
165.17	odd	20		4950.2.c.p.199.1		2
165.83	odd	20		4950.2.c.p.199.2		2
165.149	even	10		4950.2.a.bu.1.1		1
231.83	odd	10		9702.2.a.x.1.1		1
264.83	odd	10		6336.2.a.cj.1.1		1
264.149	even	10		6336.2.a.bw.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	11.6	odd	10
198.2.a.a.1.1		1	33.17	even	10
528.2.a.j.1.1		1	44.39	even	10
726.2.a.c.1.1		1	11.5	even	5
726.2.e.g.487.1		4	11.8	odd	10
726.2.e.g.493.1		4	11.7	odd	10
726.2.e.g.511.1		4	11.2	odd	10
726.2.e.g.565.1		4	11.10	odd	2
726.2.e.o.487.1		4	11.3	even	5	inner
726.2.e.o.493.1		4	11.4	even	5	inner
726.2.e.o.511.1		4	11.9	even	5	inner
726.2.e.o.565.1		4	1.1	even	1	trivial
1584.2.a.f.1.1		1	132.83	odd	10
1650.2.a.k.1.1		1	55.39	odd	10
1650.2.c.e.199.1		2	55.28	even	20
1650.2.c.e.199.2		2	55.17	even	20
1782.2.e.e.595.1		2	99.61	odd	30
1782.2.e.e.1189.1		2	99.94	odd	30
1782.2.e.v.595.1		2	99.83	even	30
1782.2.e.v.1189.1		2	99.50	even	30
2112.2.a.e.1.1		1	88.83	even	10
2112.2.a.r.1.1		1	88.61	odd	10
2178.2.a.g.1.1		1	33.5	odd	10
3234.2.a.t.1.1		1	77.6	even	10
4950.2.a.bu.1.1		1	165.149	even	10
4950.2.c.p.199.1		2	165.17	odd	20
4950.2.c.p.199.2		2	165.83	odd	20
5808.2.a.bc.1.1		1	44.27	odd	10
6336.2.a.bw.1.1		1	264.149	even	10
6336.2.a.cj.1.1		1	264.83	odd	10
9702.2.a.x.1.1		1	231.83	odd	10