Properties

Label 2028.1.o.b
Level $2028$
Weight $1$
Character orbit 2028.o
Analytic conductor $1.012$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2028.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.01210384562\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 156)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.160398576.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{3} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{4} q^{9} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{21} + q^{25} - q^{27} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{31} + \zeta_{12}^{4} q^{43} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{49} + \zeta_{12}^{4} q^{61} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{63} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{67} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{73} + \zeta_{12}^{2} q^{75} + q^{79} -\zeta_{12}^{2} q^{81} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{93} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{9} + 4q^{25} - 4q^{27} - 2q^{43} - 4q^{49} - 2q^{61} + 2q^{75} + 4q^{79} - 2q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
653.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0.500000 + 0.866025i 0 0 0 −0.866025 + 1.50000i 0 −0.500000 + 0.866025i 0
653.2 0 0.500000 + 0.866025i 0 0 0 0.866025 1.50000i 0 −0.500000 + 0.866025i 0
1205.1 0 0.500000 0.866025i 0 0 0 −0.866025 1.50000i 0 −0.500000 0.866025i 0
1205.2 0 0.500000 0.866025i 0 0 0 0.866025 + 1.50000i 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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
13.c even 3 1 inner
13.e even 6 1 inner
39.d odd 2 1 inner
39.h odd 6 1 inner
39.i odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 3 T_{7}^{2} + 9 \) acting on \(S_{1}^{\mathrm{new}}(2028, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 9 + 3 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -3 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 1 + T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( 9 + 3 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -3 + T^{2} )^{2} \)
$79$ \( ( -1 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 9 + 3 T^{2} + T^{4} \)
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