Newform orbit 6084.2.b.b

• Label: 6084.2.b.b
• Level: $6084$
• Weight: $2$
• Character orbit: 6084.b
• 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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(48.5809845897\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2*i * q^5 + i * q^7 - 2*i * q^11 - 4 * q^17 - 4*i * q^19 + 6 * q^23 + q^25 - 6 * q^29 + i * q^31 - 2 * q^35 + 10*i * q^37 - 4*i * q^41 - q^43 + 10*i * q^47 + 6 * q^49 - 8 * q^53 + 4 * q^55 + 2*i * q^59 - 5 * q^61 + 7*i * q^67 + 10*i * q^71 - 7*i * q^73 + 2 * q^77 + 17 * q^79 + 12*i * q^83 - 8*i * q^85 + 16*i * q^89 + 8 * q^95 - 13*i * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{17} + 12 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{35} - 2 q^{43} + 12 q^{49} - 16 q^{53} + 8 q^{55} - 10 q^{61} + 4 q^{77} + 34 q^{79} + 16 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6084\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(3043\)	\(3889\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

4393.1

	−	1.00000i
		1.00000i

0

0

0

−

2.00000i

0

−

1.00000i

0

0

0

4393.2

0

0

0

2.00000i

0

1.00000i

0

0

0

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.b.b		2
3.b	odd	2	1		2028.2.b.c		2
13.b	even	2	1	inner	6084.2.b.b		2
13.d	odd	4	1		6084.2.a.e		1
13.d	odd	4	1		6084.2.a.l		1
13.f	odd	12	2		468.2.l.b		2
39.d	odd	2	1		2028.2.b.c		2
39.f	even	4	1		2028.2.a.a		1
39.f	even	4	1		2028.2.a.b		1
39.h	odd	6	2		2028.2.q.g		4
39.i	odd	6	2		2028.2.q.g		4
39.k	even	12	2		156.2.i.a	✓	2
39.k	even	12	2		2028.2.i.f		2
52.l	even	12	2		1872.2.t.e		2
156.l	odd	4	1		8112.2.a.u		1
156.l	odd	4	1		8112.2.a.bd		1
156.v	odd	12	2		624.2.q.d		2
195.bc	odd	12	2		3900.2.by.c		4
195.bh	even	12	2		3900.2.q.e		2
195.bn	odd	12	2		3900.2.by.c		4

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6084, [\chi])\):

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

\( T_{7}^{2} + 1 \)

\( T_{11}^{2} + 4 \)

\( T_{23} - 6 \)

Hecke characteristic polynomials

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