Embedded newform 726.2.a.i.1.1

• Label: 726.2.a.i.1.1
• Level: $726$
• Weight: $2$
• Character: 726.1
• Self dual: yes
• Analytic conductor: $5.797$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	726.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.79713918674\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	726.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	1.00000	0.577350
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
			−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	0	0
			\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
0.554700	+	0.832050i				\(0.312833\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	−10.0000	−1.64399	−0.821995	−	0.569495i	\(-0.807139\pi\)
			−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	726.2.a.i.1.1		1
3.2	odd	2		2178.2.a.b.1.1		1
4.3	odd	2		5808.2.a.l.1.1		1
11.2	odd	10		726.2.e.k.565.1		4
11.3	even	5		726.2.e.b.493.1		4
11.4	even	5		726.2.e.b.511.1		4
11.5	even	5		726.2.e.b.487.1		4
11.6	odd	10		726.2.e.k.487.1		4
11.7	odd	10		726.2.e.k.511.1		4
11.8	odd	10		726.2.e.k.493.1		4
11.9	even	5		726.2.e.b.565.1		4
11.10	odd	2		66.2.a.a.1.1	✓	1
33.32	even	2		198.2.a.e.1.1		1
44.43	even	2		528.2.a.d.1.1		1
55.32	even	4		1650.2.c.d.199.1		2
55.43	even	4		1650.2.c.d.199.2		2
55.54	odd	2		1650.2.a.m.1.1		1
77.76	even	2		3234.2.a.d.1.1		1
88.21	odd	2		2112.2.a.i.1.1		1
88.43	even	2		2112.2.a.v.1.1		1
99.32	even	6		1782.2.e.f.1189.1		2
99.43	odd	6		1782.2.e.s.595.1		2
99.65	even	6		1782.2.e.f.595.1		2
99.76	odd	6		1782.2.e.s.1189.1		2
132.131	odd	2		1584.2.a.h.1.1		1
165.32	odd	4		4950.2.c.r.199.2		2
165.98	odd	4		4950.2.c.r.199.1		2
165.164	even	2		4950.2.a.g.1.1		1
231.230	odd	2		9702.2.a.bu.1.1		1
264.131	odd	2		6336.2.a.bf.1.1		1
264.197	even	2		6336.2.a.bj.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.a.1.1	✓	1	11.10	odd	2
198.2.a.e.1.1		1	33.32	even	2
528.2.a.d.1.1		1	44.43	even	2
726.2.a.i.1.1		1	1.1	even	1	trivial
726.2.e.b.487.1		4	11.5	even	5
726.2.e.b.493.1		4	11.3	even	5
726.2.e.b.511.1		4	11.4	even	5
726.2.e.b.565.1		4	11.9	even	5
726.2.e.k.487.1		4	11.6	odd	10
726.2.e.k.493.1		4	11.8	odd	10
726.2.e.k.511.1		4	11.7	odd	10
726.2.e.k.565.1		4	11.2	odd	10
1584.2.a.h.1.1		1	132.131	odd	2
1650.2.a.m.1.1		1	55.54	odd	2
1650.2.c.d.199.1		2	55.32	even	4
1650.2.c.d.199.2		2	55.43	even	4
1782.2.e.f.595.1		2	99.65	even	6
1782.2.e.f.1189.1		2	99.32	even	6
1782.2.e.s.595.1		2	99.43	odd	6
1782.2.e.s.1189.1		2	99.76	odd	6
2112.2.a.i.1.1		1	88.21	odd	2
2112.2.a.v.1.1		1	88.43	even	2
2178.2.a.b.1.1		1	3.2	odd	2
3234.2.a.d.1.1		1	77.76	even	2
4950.2.a.g.1.1		1	165.164	even	2
4950.2.c.r.199.1		2	165.98	odd	4
4950.2.c.r.199.2		2	165.32	odd	4
5808.2.a.l.1.1		1	4.3	odd	2
6336.2.a.bf.1.1		1	264.131	odd	2
6336.2.a.bj.1.1		1	264.197	even	2
9702.2.a.bu.1.1		1	231.230	odd	2