Newform orbit 2028.1.d.c

• Label: 2028.1.d.c
• Level: $2028$
• Weight: $1$
• Character orbit: 2028.d
• Self dual: yes
• Analytic conductor: $1.012$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2028.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.01210384562\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.160398576.1

$q$-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.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

677.1

1.73205
−1.73205

0

−1.00000

0

0

0

−1.73205

0

1.00000

0

677.2

0

−1.00000

0

0

0

1.73205

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
13.b	even	2	1	inner
39.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2028.1.d.c		2
3.b	odd	2	1	CM	2028.1.d.c		2
13.b	even	2	1	inner	2028.1.d.c		2
13.c	even	3	2		2028.1.o.b		4
13.d	odd	4	2		2028.1.g.a		2
13.e	even	6	2		2028.1.o.b		4
13.f	odd	12	2		156.1.s.a	✓	2
13.f	odd	12	2		2028.1.s.a		2
39.d	odd	2	1	inner	2028.1.d.c		2
39.f	even	4	2		2028.1.g.a		2
39.h	odd	6	2		2028.1.o.b		4
39.i	odd	6	2		2028.1.o.b		4
39.k	even	12	2		156.1.s.a	✓	2
39.k	even	12	2		2028.1.s.a		2
52.l	even	12	2		624.1.cb.a		2
65.o	even	12	2		3900.1.br.b		4
65.s	odd	12	2		3900.1.ca.b		2
65.t	even	12	2		3900.1.br.b		4
104.u	even	12	2		2496.1.cb.b		2
104.x	odd	12	2		2496.1.cb.a		2
156.v	odd	12	2		624.1.cb.a		2
195.bc	odd	12	2		3900.1.br.b		4
195.bh	even	12	2		3900.1.ca.b		2
195.bn	odd	12	2		3900.1.br.b		4
312.bo	even	12	2		2496.1.cb.a		2
312.bq	odd	12	2		2496.1.cb.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.1.s.a	✓	2	13.f	odd	12	2
156.1.s.a	✓	2	39.k	even	12	2
624.1.cb.a		2	52.l	even	12	2
624.1.cb.a		2	156.v	odd	12	2
2028.1.d.c		2	1.a	even	1	1	trivial
2028.1.d.c		2	3.b	odd	2	1	CM
2028.1.d.c		2	13.b	even	2	1	inner
2028.1.d.c		2	39.d	odd	2	1	inner
2028.1.g.a		2	13.d	odd	4	2
2028.1.g.a		2	39.f	even	4	2
2028.1.o.b		4	13.c	even	3	2
2028.1.o.b		4	13.e	even	6	2
2028.1.o.b		4	39.h	odd	6	2
2028.1.o.b		4	39.i	odd	6	2
2028.1.s.a		2	13.f	odd	12	2
2028.1.s.a		2	39.k	even	12	2
2496.1.cb.a		2	104.x	odd	12	2
2496.1.cb.a		2	312.bo	even	12	2
2496.1.cb.b		2	104.u	even	12	2
2496.1.cb.b		2	312.bq	odd	12	2
3900.1.br.b		4	65.o	even	12	2
3900.1.br.b		4	65.t	even	12	2
3900.1.br.b		4	195.bc	odd	12	2
3900.1.br.b		4	195.bn	odd	12	2
3900.1.ca.b		2	65.s	odd	12	2
3900.1.ca.b		2	195.bh	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(2028, [\chi])\).

Hecke characteristic polynomials

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