Newform orbit 2028.2.b.b

• Label: 2028.2.b.b
• Level: $2028$
• Weight: $2$
• Character orbit: 2028.b
• Analytic conductor: $16.194$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.1936615299\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{3} - \beta q^{5} - 2 \beta q^{7} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(1015\)	\(1861\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.00000

0

−

1.73205i

0

−

3.46410i

0

1.00000

0

337.2

0

−1.00000

0

1.73205i

0

3.46410i

0

1.00000

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	2028.2.b.b		2
3.b	odd	2	1		6084.2.b.c		2
13.b	even	2	1	inner	2028.2.b.b		2
13.c	even	3	1		156.2.q.a	✓	2
13.c	even	3	1		2028.2.q.a		2
13.d	odd	4	2		2028.2.a.h		2
13.e	even	6	1		156.2.q.a	✓	2
13.e	even	6	1		2028.2.q.a		2
13.f	odd	12	4		2028.2.i.h		4
39.d	odd	2	1		6084.2.b.c		2
39.f	even	4	2		6084.2.a.u		2
39.h	odd	6	1		468.2.t.c		2
39.i	odd	6	1		468.2.t.c		2
52.f	even	4	2		8112.2.a.bt		2
52.i	odd	6	1		624.2.bv.a		2
52.j	odd	6	1		624.2.bv.a		2
65.l	even	6	1		3900.2.cd.a		2
65.n	even	6	1		3900.2.cd.a		2
65.q	odd	12	2		3900.2.bw.e		4
65.r	odd	12	2		3900.2.bw.e		4
156.p	even	6	1		1872.2.by.b		2
156.r	even	6	1		1872.2.by.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.q.a	✓	2	13.c	even	3	1
156.2.q.a	✓	2	13.e	even	6	1
468.2.t.c		2	39.h	odd	6	1
468.2.t.c		2	39.i	odd	6	1
624.2.bv.a		2	52.i	odd	6	1
624.2.bv.a		2	52.j	odd	6	1
1872.2.by.b		2	156.p	even	6	1
1872.2.by.b		2	156.r	even	6	1
2028.2.a.h		2	13.d	odd	4	2
2028.2.b.b		2	1.a	even	1	1	trivial
2028.2.b.b		2	13.b	even	2	1	inner
2028.2.i.h		4	13.f	odd	12	4
2028.2.q.a		2	13.c	even	3	1
2028.2.q.a		2	13.e	even	6	1
3900.2.bw.e		4	65.q	odd	12	2
3900.2.bw.e		4	65.r	odd	12	2
3900.2.cd.a		2	65.l	even	6	1
3900.2.cd.a		2	65.n	even	6	1
6084.2.a.u		2	39.f	even	4	2
6084.2.b.c		2	3.b	odd	2	1
6084.2.b.c		2	39.d	odd	2	1
8112.2.a.bt		2	52.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(2028, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + 3 \)
$7$	\( T^{2} + 12 \)
$11$	\( T^{2} + 12 \)