Newform orbit 3042.2.a.a

• Label: 3042.2.a.a
• Level: $3042$
• Weight: $2$
• Character orbit: 3042.a
• 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:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

−1.00000

0

1.00000

−3.00000

0

1.00000

−1.00000

0

3.00000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(13\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3042.2.a.a		1
3.b	odd	2	1		338.2.a.f		1
12.b	even	2	1		2704.2.a.f		1
13.b	even	2	1		234.2.a.e		1
13.d	odd	4	2		3042.2.b.a		2
15.d	odd	2	1		8450.2.a.c		1
39.d	odd	2	1		26.2.a.a	✓	1
39.f	even	4	2		338.2.b.c		2
39.h	odd	6	2		338.2.c.d		2
39.i	odd	6	2		338.2.c.a		2
39.k	even	12	4		338.2.e.a		4
52.b	odd	2	1		1872.2.a.q		1
65.d	even	2	1		5850.2.a.p		1
65.h	odd	4	2		5850.2.e.a		2
104.e	even	2	1		7488.2.a.g		1
104.h	odd	2	1		7488.2.a.h		1
117.n	odd	6	2		2106.2.e.ba		2
117.t	even	6	2		2106.2.e.b		2
156.h	even	2	1		208.2.a.a		1
156.l	odd	4	2		2704.2.f.d		2
195.e	odd	2	1		650.2.a.j		1
195.s	even	4	2		650.2.b.d		2
273.g	even	2	1		1274.2.a.d		1
273.w	odd	6	2		1274.2.f.p		2
273.ba	even	6	2		1274.2.f.r		2
312.b	odd	2	1		832.2.a.d		1
312.h	even	2	1		832.2.a.i		1
429.e	even	2	1		3146.2.a.n		1
624.v	even	4	2		3328.2.b.j		2
624.bi	odd	4	2		3328.2.b.m		2
663.g	odd	2	1		7514.2.a.c		1
741.d	even	2	1		9386.2.a.j		1
780.d	even	2	1		5200.2.a.x		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.a.a	✓	1	39.d	odd	2	1
208.2.a.a		1	156.h	even	2	1
234.2.a.e		1	13.b	even	2	1
338.2.a.f		1	3.b	odd	2	1
338.2.b.c		2	39.f	even	4	2
338.2.c.a		2	39.i	odd	6	2
338.2.c.d		2	39.h	odd	6	2
338.2.e.a		4	39.k	even	12	4
650.2.a.j		1	195.e	odd	2	1
650.2.b.d		2	195.s	even	4	2
832.2.a.d		1	312.b	odd	2	1
832.2.a.i		1	312.h	even	2	1
1274.2.a.d		1	273.g	even	2	1
1274.2.f.p		2	273.w	odd	6	2
1274.2.f.r		2	273.ba	even	6	2
1872.2.a.q		1	52.b	odd	2	1
2106.2.e.b		2	117.t	even	6	2
2106.2.e.ba		2	117.n	odd	6	2
2704.2.a.f		1	12.b	even	2	1
2704.2.f.d		2	156.l	odd	4	2
3042.2.a.a		1	1.a	even	1	1	trivial
3042.2.b.a		2	13.d	odd	4	2
3146.2.a.n		1	429.e	even	2	1
3328.2.b.j		2	624.v	even	4	2
3328.2.b.m		2	624.bi	odd	4	2
5200.2.a.x		1	780.d	even	2	1
5850.2.a.p		1	65.d	even	2	1
5850.2.e.a		2	65.h	odd	4	2
7488.2.a.g		1	104.e	even	2	1
7488.2.a.h		1	104.h	odd	2	1
7514.2.a.c		1	663.g	odd	2	1
8450.2.a.c		1	15.d	odd	2	1
9386.2.a.j		1	741.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\):

\( T_{5} + 3 \)

\( T_{7} - 1 \)

\( T_{11} - 6 \)

\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1 \)
$3$	\( T \)
$5$	\( T + 3 \)
$7$	\( T - 1 \)
$11$	\( T - 6 \)