Embedded newform 2704.2.a.h.1.1

• Label: 2704.2.a.h.1.1
• Level: $2704$
• Weight: $2$
• Character: 2704.1
• Self dual: yes
• Analytic conductor: $21.592$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(21.5915487066\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2704.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)

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
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	−1.00000	−0.447214	−0.223607	−	0.974679i	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	0	0
			\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
0.363803	+	0.931476i				\(0.381478\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−1.00000	−0.185695	−0.0928477	−	0.995680i	\(-0.529597\pi\)
			−0.0928477	+	0.995680i	\(0.529597\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	−9.00000	−1.40556	−0.702782	−	0.711405i	\(-0.748059\pi\)
			−0.702782	+	0.711405i	\(0.748059\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.h.1.1		1
4.3	odd	2		338.2.a.e.1.1		1
12.11	even	2		3042.2.a.e.1.1		1
13.3	even	3		208.2.i.b.113.1		2
13.5	odd	4		2704.2.f.g.337.2		2
13.8	odd	4		2704.2.f.g.337.1		2
13.9	even	3		208.2.i.b.81.1		2
13.12	even	2		2704.2.a.i.1.1		1
20.19	odd	2		8450.2.a.f.1.1		1
39.29	odd	6		1872.2.t.k.1153.1		2
39.35	odd	6		1872.2.t.k.289.1		2
52.3	odd	6		26.2.c.a.9.1	yes	2
52.7	even	12		338.2.e.b.23.2		4
52.11	even	12		338.2.e.b.147.1		4
52.15	even	12		338.2.e.b.147.2		4
52.19	even	12		338.2.e.b.23.1		4
52.23	odd	6		338.2.c.e.191.1		2
52.31	even	4		338.2.b.b.337.1		2
52.35	odd	6		26.2.c.a.3.1	✓	2
52.43	odd	6		338.2.c.e.315.1		2
52.47	even	4		338.2.b.b.337.2		2
52.51	odd	2		338.2.a.c.1.1		1
104.3	odd	6		832.2.i.e.321.1		2
104.29	even	6		832.2.i.f.321.1		2
104.35	odd	6		832.2.i.e.705.1		2
104.61	even	6		832.2.i.f.705.1		2
156.35	even	6		234.2.h.c.55.1		2
156.47	odd	4		3042.2.b.e.1351.1		2
156.83	odd	4		3042.2.b.e.1351.2		2
156.107	even	6		234.2.h.c.217.1		2
156.155	even	2		3042.2.a.k.1.1		1
260.3	even	12		650.2.o.c.399.2		4
260.87	even	12		650.2.o.c.549.2		4
260.107	even	12		650.2.o.c.399.1		4
260.139	odd	6		650.2.e.c.601.1		2
260.159	odd	6		650.2.e.c.451.1		2
260.243	even	12		650.2.o.c.549.1		4
260.259	odd	2		8450.2.a.s.1.1		1
364.3	even	6		1274.2.h.a.373.1		2
364.55	even	6		1274.2.g.a.295.1		2
364.87	even	6		1274.2.e.m.471.1		2
364.107	odd	6		1274.2.e.n.165.1		2
364.139	even	6		1274.2.g.a.393.1		2
364.159	even	6		1274.2.e.m.165.1		2
364.191	odd	6		1274.2.h.b.263.1		2
364.243	even	6		1274.2.h.a.263.1		2
364.263	odd	6		1274.2.h.b.373.1		2
364.347	odd	6		1274.2.e.n.471.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.c.a.3.1	✓	2	52.35	odd	6
26.2.c.a.9.1	yes	2	52.3	odd	6
208.2.i.b.81.1		2	13.9	even	3
208.2.i.b.113.1		2	13.3	even	3
234.2.h.c.55.1		2	156.35	even	6
234.2.h.c.217.1		2	156.107	even	6
338.2.a.c.1.1		1	52.51	odd	2
338.2.a.e.1.1		1	4.3	odd	2
338.2.b.b.337.1		2	52.31	even	4
338.2.b.b.337.2		2	52.47	even	4
338.2.c.e.191.1		2	52.23	odd	6
338.2.c.e.315.1		2	52.43	odd	6
338.2.e.b.23.1		4	52.19	even	12
338.2.e.b.23.2		4	52.7	even	12
338.2.e.b.147.1		4	52.11	even	12
338.2.e.b.147.2		4	52.15	even	12
650.2.e.c.451.1		2	260.159	odd	6
650.2.e.c.601.1		2	260.139	odd	6
650.2.o.c.399.1		4	260.107	even	12
650.2.o.c.399.2		4	260.3	even	12
650.2.o.c.549.1		4	260.243	even	12
650.2.o.c.549.2		4	260.87	even	12
832.2.i.e.321.1		2	104.3	odd	6
832.2.i.e.705.1		2	104.35	odd	6
832.2.i.f.321.1		2	104.29	even	6
832.2.i.f.705.1		2	104.61	even	6
1274.2.e.m.165.1		2	364.159	even	6
1274.2.e.m.471.1		2	364.87	even	6
1274.2.e.n.165.1		2	364.107	odd	6
1274.2.e.n.471.1		2	364.347	odd	6
1274.2.g.a.295.1		2	364.55	even	6
1274.2.g.a.393.1		2	364.139	even	6
1274.2.h.a.263.1		2	364.243	even	6
1274.2.h.a.373.1		2	364.3	even	6
1274.2.h.b.263.1		2	364.191	odd	6
1274.2.h.b.373.1		2	364.263	odd	6
1872.2.t.k.289.1		2	39.35	odd	6
1872.2.t.k.1153.1		2	39.29	odd	6
2704.2.a.h.1.1		1	1.1	even	1	trivial
2704.2.a.i.1.1		1	13.12	even	2
2704.2.f.g.337.1		2	13.8	odd	4
2704.2.f.g.337.2		2	13.5	odd	4
3042.2.a.e.1.1		1	12.11	even	2
3042.2.a.k.1.1		1	156.155	even	2
3042.2.b.e.1351.1		2	156.47	odd	4
3042.2.b.e.1351.2		2	156.83	odd	4
8450.2.a.f.1.1		1	20.19	odd	2
8450.2.a.s.1.1		1	260.259	odd	2