Newform orbit 3042.2.a.y

• Label: 3042.2.a.y
• Level: $3042$
• Weight: $2$
• Character orbit: 3042.a
• Self dual: yes
• Analytic conductor: $24.290$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + (\beta + 2) q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \)

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

−1.73205
1.73205

1.00000

0

1.00000

0.267949

0

−0.732051

1.00000

0

0.267949

1.2

1.00000

0

1.00000

3.73205

0

2.73205

1.00000

0

3.73205

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.y		2
3.b	odd	2	1		1014.2.a.i		2
12.b	even	2	1		8112.2.a.bj		2
13.b	even	2	1		3042.2.a.p		2
13.d	odd	4	2		3042.2.b.i		4
13.f	odd	12	2		234.2.l.c		4
39.d	odd	2	1		1014.2.a.k		2
39.f	even	4	2		1014.2.b.e		4
39.h	odd	6	2		1014.2.e.g		4
39.i	odd	6	2		1014.2.e.i		4
39.k	even	12	2		78.2.i.a	✓	4
39.k	even	12	2		1014.2.i.a		4
52.l	even	12	2		1872.2.by.h		4
156.h	even	2	1		8112.2.a.bp		2
156.v	odd	12	2		624.2.bv.e		4
195.bc	odd	12	2		1950.2.y.b		4
195.bh	even	12	2		1950.2.bc.d		4
195.bn	odd	12	2		1950.2.y.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.i.a	✓	4	39.k	even	12	2
234.2.l.c		4	13.f	odd	12	2
624.2.bv.e		4	156.v	odd	12	2
1014.2.a.i		2	3.b	odd	2	1
1014.2.a.k		2	39.d	odd	2	1
1014.2.b.e		4	39.f	even	4	2
1014.2.e.g		4	39.h	odd	6	2
1014.2.e.i		4	39.i	odd	6	2
1014.2.i.a		4	39.k	even	12	2
1872.2.by.h		4	52.l	even	12	2
1950.2.y.b		4	195.bc	odd	12	2
1950.2.y.g		4	195.bn	odd	12	2
1950.2.bc.d		4	195.bh	even	12	2
3042.2.a.p		2	13.b	even	2	1
3042.2.a.y		2	1.a	even	1	1	trivial
3042.2.b.i		4	13.d	odd	4	2
8112.2.a.bj		2	12.b	even	2	1
8112.2.a.bp		2	156.h	even	2	1

Hecke kernels

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

\( T_{5}^{2} - 4T_{5} + 1 \)

\( T_{7}^{2} - 2T_{7} - 2 \)

\( T_{11}^{2} - 6T_{11} + 6 \)

\( T_{17}^{2} - 8T_{17} + 13 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 4T + 1 \)
$7$	\( T^{2} - 2T - 2 \)
$11$	\( T^{2} - 6T + 6 \)