Embedded newform 3042.2.a.l.1.1

• Label: 3042.2.a.l.1.1
• Level: $3042$
• Weight: $2$
• Character: 3042.1
• Self dual: yes
• Analytic conductor: $24.290$
• Analytic rank: $1$
• 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:	\(1\)
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

\(f(q)\)

\(=\)

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

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	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	−1.00000	−0.377964	−0.188982	−	0.981981i	\(-0.560519\pi\)
			−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	−2.00000	−0.603023	−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	0	0
			\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
0.363803	+	0.931476i				\(0.381478\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	−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	\(-0.579313\pi\)
			−0.246598	+	0.969118i	\(0.579313\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−5.00000	−0.762493	−0.381246	−	0.924473i	\(-0.624505\pi\)
			−0.381246	+	0.924473i	\(0.624505\pi\)
\(47\)	13.0000	1.89624	0.948122	−	0.317905i	\(-0.102979\pi\)
			0.948122	+	0.317905i	\(0.102979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.l.1.1		1
3.2	odd	2		338.2.a.a.1.1		1
12.11	even	2		2704.2.a.n.1.1		1
13.5	odd	4		3042.2.b.f.1351.1		2
13.8	odd	4		3042.2.b.f.1351.2		2
13.12	even	2		234.2.a.b.1.1		1
15.14	odd	2		8450.2.a.y.1.1		1
39.2	even	12		338.2.e.d.147.1		4
39.5	even	4		338.2.b.a.337.2		2
39.8	even	4		338.2.b.a.337.1		2
39.11	even	12		338.2.e.d.147.2		4
39.17	odd	6		338.2.c.c.315.1		2
39.20	even	12		338.2.e.d.23.1		4
39.23	odd	6		338.2.c.c.191.1		2
39.29	odd	6		338.2.c.g.191.1		2
39.32	even	12		338.2.e.d.23.2		4
39.35	odd	6		338.2.c.g.315.1		2
39.38	odd	2		26.2.a.b.1.1	✓	1
52.51	odd	2		1872.2.a.m.1.1		1
65.12	odd	4		5850.2.e.v.5149.1		2
65.38	odd	4		5850.2.e.v.5149.2		2
65.64	even	2		5850.2.a.bn.1.1		1
104.51	odd	2		7488.2.a.v.1.1		1
104.77	even	2		7488.2.a.w.1.1		1
117.25	even	6		2106.2.e.t.1405.1		2
117.38	odd	6		2106.2.e.h.1405.1		2
117.77	odd	6		2106.2.e.h.703.1		2
117.103	even	6		2106.2.e.t.703.1		2
156.47	odd	4		2704.2.f.j.337.2		2
156.83	odd	4		2704.2.f.j.337.1		2
156.155	even	2		208.2.a.d.1.1		1
195.38	even	4		650.2.b.a.599.1		2
195.77	even	4		650.2.b.a.599.2		2
195.194	odd	2		650.2.a.g.1.1		1
273.38	even	6		1274.2.f.a.79.1		2
273.116	odd	6		1274.2.f.l.79.1		2
273.194	even	6		1274.2.f.a.1145.1		2
273.233	odd	6		1274.2.f.l.1145.1		2
273.272	even	2		1274.2.a.o.1.1		1
312.77	odd	2		832.2.a.j.1.1		1
312.155	even	2		832.2.a.a.1.1		1
429.428	even	2		3146.2.a.a.1.1		1
624.77	odd	4		3328.2.b.g.1665.1		2
624.155	even	4		3328.2.b.k.1665.1		2
624.389	odd	4		3328.2.b.g.1665.2		2
624.467	even	4		3328.2.b.k.1665.2		2
663.662	odd	2		7514.2.a.i.1.1		1
741.740	even	2		9386.2.a.f.1.1		1
780.779	even	2		5200.2.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.b.1.1	✓	1	39.38	odd	2
208.2.a.d.1.1		1	156.155	even	2
234.2.a.b.1.1		1	13.12	even	2
338.2.a.a.1.1		1	3.2	odd	2
338.2.b.a.337.1		2	39.8	even	4
338.2.b.a.337.2		2	39.5	even	4
338.2.c.c.191.1		2	39.23	odd	6
338.2.c.c.315.1		2	39.17	odd	6
338.2.c.g.191.1		2	39.29	odd	6
338.2.c.g.315.1		2	39.35	odd	6
338.2.e.d.23.1		4	39.20	even	12
338.2.e.d.23.2		4	39.32	even	12
338.2.e.d.147.1		4	39.2	even	12
338.2.e.d.147.2		4	39.11	even	12
650.2.a.g.1.1		1	195.194	odd	2
650.2.b.a.599.1		2	195.38	even	4
650.2.b.a.599.2		2	195.77	even	4
832.2.a.a.1.1		1	312.155	even	2
832.2.a.j.1.1		1	312.77	odd	2
1274.2.a.o.1.1		1	273.272	even	2
1274.2.f.a.79.1		2	273.38	even	6
1274.2.f.a.1145.1		2	273.194	even	6
1274.2.f.l.79.1		2	273.116	odd	6
1274.2.f.l.1145.1		2	273.233	odd	6
1872.2.a.m.1.1		1	52.51	odd	2
2106.2.e.h.703.1		2	117.77	odd	6
2106.2.e.h.1405.1		2	117.38	odd	6
2106.2.e.t.703.1		2	117.103	even	6
2106.2.e.t.1405.1		2	117.25	even	6
2704.2.a.n.1.1		1	12.11	even	2
2704.2.f.j.337.1		2	156.83	odd	4
2704.2.f.j.337.2		2	156.47	odd	4
3042.2.a.l.1.1		1	1.1	even	1	trivial
3042.2.b.f.1351.1		2	13.5	odd	4
3042.2.b.f.1351.2		2	13.8	odd	4
3146.2.a.a.1.1		1	429.428	even	2
3328.2.b.g.1665.1		2	624.77	odd	4
3328.2.b.g.1665.2		2	624.389	odd	4
3328.2.b.k.1665.1		2	624.155	even	4
3328.2.b.k.1665.2		2	624.467	even	4
5200.2.a.c.1.1		1	780.779	even	2
5850.2.a.bn.1.1		1	65.64	even	2
5850.2.e.v.5149.1		2	65.12	odd	4
5850.2.e.v.5149.2		2	65.38	odd	4
7488.2.a.v.1.1		1	104.51	odd	2
7488.2.a.w.1.1		1	104.77	even	2
7514.2.a.i.1.1		1	663.662	odd	2
8450.2.a.y.1.1		1	15.14	odd	2
9386.2.a.f.1.1		1	741.740	even	2