Newform orbit 6084.2.a.u

• Label: 6084.2.a.u
• Level: $6084$
• Weight: $2$
• Character orbit: 6084.a
• Self dual: yes
• Analytic conductor: $48.581$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(48.5809845897\)
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 156)
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 + b * q^5 + 2*b * q^7 + 2*b * q^11 + 3 * q^17 + 2*b * q^19 + 6 * q^23 - 2 * q^25 - 9 * q^29 + 6 * q^35 + 3*b * q^37 + 5*b * q^41 - 2 * q^43 + 2*b * q^47 + 5 * q^49 + 9 * q^53 + 6 * q^55 - 8*b * q^59 - 11 * q^61 + 6*b * q^67 + 6*b * q^71 - 3*b * q^73 + 12 * q^77 - 8 * q^79 + 2*b * q^83 + 3*b * q^85 - 4*b * q^89 + 6 * q^95 - 4*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{17} + 12 q^{23} - 4 q^{25} - 18 q^{29} + 12 q^{35} - 4 q^{43} + 10 q^{49} + 18 q^{53} + 12 q^{55} - 22 q^{61} + 24 q^{77} - 16 q^{79} + 12 q^{95}+O(q^{100}) \)

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

0

0

0

−1.73205

0

−3.46410

0

0

0

1.2

0

0

0

1.73205

0

3.46410

0

0

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6084.2.a.u		2
3.b	odd	2	1		2028.2.a.h		2
12.b	even	2	1		8112.2.a.bt		2
13.b	even	2	1	inner	6084.2.a.u		2
13.d	odd	4	2		6084.2.b.c		2
13.f	odd	12	2		468.2.t.c		2
39.d	odd	2	1		2028.2.a.h		2
39.f	even	4	2		2028.2.b.b		2
39.h	odd	6	2		2028.2.i.h		4
39.i	odd	6	2		2028.2.i.h		4
39.k	even	12	2		156.2.q.a	✓	2
39.k	even	12	2		2028.2.q.a		2
52.l	even	12	2		1872.2.by.b		2
156.h	even	2	1		8112.2.a.bt		2
156.v	odd	12	2		624.2.bv.a		2
195.bc	odd	12	2		3900.2.bw.e		4
195.bh	even	12	2		3900.2.cd.a		2
195.bn	odd	12	2		3900.2.bw.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.q.a	✓	2	39.k	even	12	2
468.2.t.c		2	13.f	odd	12	2
624.2.bv.a		2	156.v	odd	12	2
1872.2.by.b		2	52.l	even	12	2
2028.2.a.h		2	3.b	odd	2	1
2028.2.a.h		2	39.d	odd	2	1
2028.2.b.b		2	39.f	even	4	2
2028.2.i.h		4	39.h	odd	6	2
2028.2.i.h		4	39.i	odd	6	2
2028.2.q.a		2	39.k	even	12	2
3900.2.bw.e		4	195.bc	odd	12	2
3900.2.bw.e		4	195.bn	odd	12	2
3900.2.cd.a		2	195.bh	even	12	2
6084.2.a.u		2	1.a	even	1	1	trivial
6084.2.a.u		2	13.b	even	2	1	inner
6084.2.b.c		2	13.d	odd	4	2
8112.2.a.bt		2	12.b	even	2	1
8112.2.a.bt		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6084))\):

\( T_{5}^{2} - 3 \)

\( T_{7}^{2} - 12 \)

\( T_{11}^{2} - 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3 \)
$7$	\( T^{2} - 12 \)
$11$	\( T^{2} - 12 \)