Properties

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 $4$

Related objects

Downloads

Learn more

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.

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

Character values

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

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

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
\(n\): 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
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

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