Embedded newform 3042.2.a.e.1.1

• Label: 3042.2.a.e.1.1
• Level: $3042$
• Weight: $2$
• Character: 3042.1
• Self dual: yes
• Analytic conductor: $24.290$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(24.2904922949\)
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	\(\chi\)	\(=\)	3042.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)

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
			\(3\)	0	0
\(5\)	1.00000	0.447214				0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	−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	\(-0.618522\pi\)
−0.363803	+	0.931476i				\(0.618522\pi\)
\(19\)	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	\(-0.470403\pi\)
			0.0928477	+	0.995680i	\(0.470403\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	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	\(-0.251941\pi\)
			0.702782	+	0.711405i	\(0.251941\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	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	3042.2.a.e.1.1		1
3.2	odd	2		338.2.a.e.1.1		1
12.11	even	2		2704.2.a.h.1.1		1
13.3	even	3		234.2.h.c.217.1		2
13.5	odd	4		3042.2.b.e.1351.2		2
13.8	odd	4		3042.2.b.e.1351.1		2
13.9	even	3		234.2.h.c.55.1		2
13.12	even	2		3042.2.a.k.1.1		1
15.14	odd	2		8450.2.a.f.1.1		1
39.2	even	12		338.2.e.b.147.2		4
39.5	even	4		338.2.b.b.337.1		2
39.8	even	4		338.2.b.b.337.2		2
39.11	even	12		338.2.e.b.147.1		4
39.17	odd	6		338.2.c.e.315.1		2
39.20	even	12		338.2.e.b.23.2		4
39.23	odd	6		338.2.c.e.191.1		2
39.29	odd	6		26.2.c.a.9.1	yes	2
39.32	even	12		338.2.e.b.23.1		4
39.35	odd	6		26.2.c.a.3.1	✓	2
39.38	odd	2		338.2.a.c.1.1		1
52.3	odd	6		1872.2.t.k.1153.1		2
52.35	odd	6		1872.2.t.k.289.1		2
156.35	even	6		208.2.i.b.81.1		2
156.47	odd	4		2704.2.f.g.337.1		2
156.83	odd	4		2704.2.f.g.337.2		2
156.107	even	6		208.2.i.b.113.1		2
156.155	even	2		2704.2.a.i.1.1		1
195.29	odd	6		650.2.e.c.451.1		2
195.68	even	12		650.2.o.c.399.2		4
195.74	odd	6		650.2.e.c.601.1		2
195.107	even	12		650.2.o.c.399.1		4
195.113	even	12		650.2.o.c.549.1		4
195.152	even	12		650.2.o.c.549.2		4
195.194	odd	2		8450.2.a.s.1.1		1
273.68	even	6		1274.2.e.m.165.1		2
273.74	odd	6		1274.2.e.n.471.1		2
273.107	odd	6		1274.2.e.n.165.1		2
273.146	even	6		1274.2.g.a.295.1		2
273.152	even	6		1274.2.h.a.263.1		2
273.185	even	6		1274.2.h.a.373.1		2
273.191	odd	6		1274.2.h.b.263.1		2
273.230	even	6		1274.2.g.a.393.1		2
273.263	odd	6		1274.2.h.b.373.1		2
273.269	even	6		1274.2.e.m.471.1		2
312.29	odd	6		832.2.i.e.321.1		2
312.35	even	6		832.2.i.f.705.1		2
312.107	even	6		832.2.i.f.321.1		2
312.269	odd	6		832.2.i.e.705.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.c.a.3.1	✓	2	39.35	odd	6
26.2.c.a.9.1	yes	2	39.29	odd	6
208.2.i.b.81.1		2	156.35	even	6
208.2.i.b.113.1		2	156.107	even	6
234.2.h.c.55.1		2	13.9	even	3
234.2.h.c.217.1		2	13.3	even	3
338.2.a.c.1.1		1	39.38	odd	2
338.2.a.e.1.1		1	3.2	odd	2
338.2.b.b.337.1		2	39.5	even	4
338.2.b.b.337.2		2	39.8	even	4
338.2.c.e.191.1		2	39.23	odd	6
338.2.c.e.315.1		2	39.17	odd	6
338.2.e.b.23.1		4	39.32	even	12
338.2.e.b.23.2		4	39.20	even	12
338.2.e.b.147.1		4	39.11	even	12
338.2.e.b.147.2		4	39.2	even	12
650.2.e.c.451.1		2	195.29	odd	6
650.2.e.c.601.1		2	195.74	odd	6
650.2.o.c.399.1		4	195.107	even	12
650.2.o.c.399.2		4	195.68	even	12
650.2.o.c.549.1		4	195.113	even	12
650.2.o.c.549.2		4	195.152	even	12
832.2.i.e.321.1		2	312.29	odd	6
832.2.i.e.705.1		2	312.269	odd	6
832.2.i.f.321.1		2	312.107	even	6
832.2.i.f.705.1		2	312.35	even	6
1274.2.e.m.165.1		2	273.68	even	6
1274.2.e.m.471.1		2	273.269	even	6
1274.2.e.n.165.1		2	273.107	odd	6
1274.2.e.n.471.1		2	273.74	odd	6
1274.2.g.a.295.1		2	273.146	even	6
1274.2.g.a.393.1		2	273.230	even	6
1274.2.h.a.263.1		2	273.152	even	6
1274.2.h.a.373.1		2	273.185	even	6
1274.2.h.b.263.1		2	273.191	odd	6
1274.2.h.b.373.1		2	273.263	odd	6
1872.2.t.k.289.1		2	52.35	odd	6
1872.2.t.k.1153.1		2	52.3	odd	6
2704.2.a.h.1.1		1	12.11	even	2
2704.2.a.i.1.1		1	156.155	even	2
2704.2.f.g.337.1		2	156.47	odd	4
2704.2.f.g.337.2		2	156.83	odd	4
3042.2.a.e.1.1		1	1.1	even	1	trivial
3042.2.a.k.1.1		1	13.12	even	2
3042.2.b.e.1351.1		2	13.8	odd	4
3042.2.b.e.1351.2		2	13.5	odd	4
8450.2.a.f.1.1		1	15.14	odd	2
8450.2.a.s.1.1		1	195.194	odd	2