Embedded newform 5776.2.a.n.1.1

• Label: 5776.2.a.n.1.1
• Level: $5776$
• Weight: $2$
• Character: 5776.1
• Self dual: yes
• Analytic conductor: $46.122$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(46.1215922075\)
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	\(\chi\)	\(=\)	5776.1

$q$-expansion

\(f(q)\)

\(=\)

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

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	−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	\(-0.284877\pi\)
0.625543	+	0.780189i				\(0.284877\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	2.00000	0.359211	0.179605	−	0.983739i	\(-0.442518\pi\)
			0.179605	+	0.983739i	\(0.442518\pi\)
\(37\)	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	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	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	5776.2.a.n.1.1		1
4.3	odd	2		722.2.a.d.1.1		1
12.11	even	2		6498.2.a.e.1.1		1
19.8	odd	6		304.2.i.c.273.1		2
19.12	odd	6		304.2.i.c.49.1		2
19.18	odd	2		5776.2.a.g.1.1		1
57.8	even	6		2736.2.s.m.577.1		2
57.50	even	6		2736.2.s.m.1873.1		2
76.3	even	18		722.2.e.j.389.1		6
76.7	odd	6		722.2.c.b.429.1		2
76.11	odd	6		722.2.c.b.653.1		2
76.15	even	18		722.2.e.j.415.1		6
76.23	odd	18		722.2.e.i.415.1		6
76.27	even	6		38.2.c.a.7.1	✓	2
76.31	even	6		38.2.c.a.11.1	yes	2
76.35	odd	18		722.2.e.i.389.1		6
76.43	odd	18		722.2.e.i.595.1		6
76.47	odd	18		722.2.e.i.423.1		6
76.51	even	18		722.2.e.j.245.1		6
76.55	odd	18		722.2.e.i.99.1		6
76.59	even	18		722.2.e.j.99.1		6
76.63	odd	18		722.2.e.i.245.1		6
76.67	even	18		722.2.e.j.423.1		6
76.71	even	18		722.2.e.j.595.1		6
76.75	even	2		722.2.a.c.1.1		1
152.27	even	6		1216.2.i.h.577.1		2
152.69	odd	6		1216.2.i.d.961.1		2
152.107	even	6		1216.2.i.h.961.1		2
152.141	odd	6		1216.2.i.d.577.1		2
228.107	odd	6		342.2.g.b.163.1		2
228.179	odd	6		342.2.g.b.235.1		2
228.227	odd	2		6498.2.a.s.1.1		1
380.27	odd	12		950.2.j.e.349.1		4
380.103	odd	12		950.2.j.e.349.2		4
380.107	odd	12		950.2.j.e.49.2		4
380.179	even	6		950.2.e.d.501.1		2
380.183	odd	12		950.2.j.e.49.1		4
380.259	even	6		950.2.e.d.201.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.a.7.1	✓	2	76.27	even	6
38.2.c.a.11.1	yes	2	76.31	even	6
304.2.i.c.49.1		2	19.12	odd	6
304.2.i.c.273.1		2	19.8	odd	6
342.2.g.b.163.1		2	228.107	odd	6
342.2.g.b.235.1		2	228.179	odd	6
722.2.a.c.1.1		1	76.75	even	2
722.2.a.d.1.1		1	4.3	odd	2
722.2.c.b.429.1		2	76.7	odd	6
722.2.c.b.653.1		2	76.11	odd	6
722.2.e.i.99.1		6	76.55	odd	18
722.2.e.i.245.1		6	76.63	odd	18
722.2.e.i.389.1		6	76.35	odd	18
722.2.e.i.415.1		6	76.23	odd	18
722.2.e.i.423.1		6	76.47	odd	18
722.2.e.i.595.1		6	76.43	odd	18
722.2.e.j.99.1		6	76.59	even	18
722.2.e.j.245.1		6	76.51	even	18
722.2.e.j.389.1		6	76.3	even	18
722.2.e.j.415.1		6	76.15	even	18
722.2.e.j.423.1		6	76.67	even	18
722.2.e.j.595.1		6	76.71	even	18
950.2.e.d.201.1		2	380.259	even	6
950.2.e.d.501.1		2	380.179	even	6
950.2.j.e.49.1		4	380.183	odd	12
950.2.j.e.49.2		4	380.107	odd	12
950.2.j.e.349.1		4	380.27	odd	12
950.2.j.e.349.2		4	380.103	odd	12
1216.2.i.d.577.1		2	152.141	odd	6
1216.2.i.d.961.1		2	152.69	odd	6
1216.2.i.h.577.1		2	152.27	even	6
1216.2.i.h.961.1		2	152.107	even	6
2736.2.s.m.577.1		2	57.8	even	6
2736.2.s.m.1873.1		2	57.50	even	6
5776.2.a.g.1.1		1	19.18	odd	2
5776.2.a.n.1.1		1	1.1	even	1	trivial
6498.2.a.e.1.1		1	12.11	even	2
6498.2.a.s.1.1		1	228.227	odd	2