Embedded newform 722.2.a.d.1.1

• Label: 722.2.a.d.1.1
• Level: $722$
• Weight: $2$
• Character: 722.1
• Self dual: yes
• Analytic conductor: $5.765$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.76519902594\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	722.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)

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	−0.288675	−	0.957427i	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
			0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	0	0
			\(23\)	−6.00000	−1.25109	−0.625543	−	0.780189i	\(-0.715123\pi\)
−0.625543	+	0.780189i				\(0.715123\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	10.0000	1.64399	0.821995	−	0.569495i	\(-0.192861\pi\)
			0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	−9.00000	−1.40556	−0.702782	−	0.711405i	\(-0.748059\pi\)
			−0.702782	+	0.711405i	\(0.748059\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.a.d.1.1		1
3.2	odd	2		6498.2.a.e.1.1		1
4.3	odd	2		5776.2.a.n.1.1		1
19.2	odd	18		722.2.e.j.99.1		6
19.3	odd	18		722.2.e.j.389.1		6
19.4	even	9		722.2.e.i.415.1		6
19.5	even	9		722.2.e.i.595.1		6
19.6	even	9		722.2.e.i.245.1		6
19.7	even	3		722.2.c.b.429.1		2
19.8	odd	6		38.2.c.a.7.1	✓	2
19.9	even	9		722.2.e.i.423.1		6
19.10	odd	18		722.2.e.j.423.1		6
19.11	even	3		722.2.c.b.653.1		2
19.12	odd	6		38.2.c.a.11.1	yes	2
19.13	odd	18		722.2.e.j.245.1		6
19.14	odd	18		722.2.e.j.595.1		6
19.15	odd	18		722.2.e.j.415.1		6
19.16	even	9		722.2.e.i.389.1		6
19.17	even	9		722.2.e.i.99.1		6
19.18	odd	2		722.2.a.c.1.1		1
57.8	even	6		342.2.g.b.235.1		2
57.50	even	6		342.2.g.b.163.1		2
57.56	even	2		6498.2.a.s.1.1		1
76.27	even	6		304.2.i.c.273.1		2
76.31	even	6		304.2.i.c.49.1		2
76.75	even	2		5776.2.a.g.1.1		1
95.8	even	12		950.2.j.e.349.2		4
95.12	even	12		950.2.j.e.49.2		4
95.27	even	12		950.2.j.e.349.1		4
95.69	odd	6		950.2.e.d.201.1		2
95.84	odd	6		950.2.e.d.501.1		2
95.88	even	12		950.2.j.e.49.1		4
152.27	even	6		1216.2.i.d.577.1		2
152.69	odd	6		1216.2.i.h.961.1		2
152.107	even	6		1216.2.i.d.961.1		2
152.141	odd	6		1216.2.i.h.577.1		2
228.107	odd	6		2736.2.s.m.1873.1		2
228.179	odd	6		2736.2.s.m.577.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.a.7.1	✓	2	19.8	odd	6
38.2.c.a.11.1	yes	2	19.12	odd	6
304.2.i.c.49.1		2	76.31	even	6
304.2.i.c.273.1		2	76.27	even	6
342.2.g.b.163.1		2	57.50	even	6
342.2.g.b.235.1		2	57.8	even	6
722.2.a.c.1.1		1	19.18	odd	2
722.2.a.d.1.1		1	1.1	even	1	trivial
722.2.c.b.429.1		2	19.7	even	3
722.2.c.b.653.1		2	19.11	even	3
722.2.e.i.99.1		6	19.17	even	9
722.2.e.i.245.1		6	19.6	even	9
722.2.e.i.389.1		6	19.16	even	9
722.2.e.i.415.1		6	19.4	even	9
722.2.e.i.423.1		6	19.9	even	9
722.2.e.i.595.1		6	19.5	even	9
722.2.e.j.99.1		6	19.2	odd	18
722.2.e.j.245.1		6	19.13	odd	18
722.2.e.j.389.1		6	19.3	odd	18
722.2.e.j.415.1		6	19.15	odd	18
722.2.e.j.423.1		6	19.10	odd	18
722.2.e.j.595.1		6	19.14	odd	18
950.2.e.d.201.1		2	95.69	odd	6
950.2.e.d.501.1		2	95.84	odd	6
950.2.j.e.49.1		4	95.88	even	12
950.2.j.e.49.2		4	95.12	even	12
950.2.j.e.349.1		4	95.27	even	12
950.2.j.e.349.2		4	95.8	even	12
1216.2.i.d.577.1		2	152.27	even	6
1216.2.i.d.961.1		2	152.107	even	6
1216.2.i.h.577.1		2	152.141	odd	6
1216.2.i.h.961.1		2	152.69	odd	6
2736.2.s.m.577.1		2	228.179	odd	6
2736.2.s.m.1873.1		2	228.107	odd	6
5776.2.a.g.1.1		1	76.75	even	2
5776.2.a.n.1.1		1	4.3	odd	2
6498.2.a.e.1.1		1	3.2	odd	2
6498.2.a.s.1.1		1	57.56	even	2