Properties

Label 2020.1.d.c
Level $2020$
Weight $1$
Character orbit 2020.d
Analytic conductor $1.008$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -20, -404, 505
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2020,1,Mod(2019,2020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2020.2019");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2020 = 2^{2} \cdot 5 \cdot 101 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2020.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00811132552\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-101})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.1632160000.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + i q^{2} - i q^{3} - q^{4} - q^{5} + 2 q^{6} - i q^{7} - i q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - i q^{3} - q^{4} - q^{5} + 2 q^{6} - i q^{7} - i q^{8} - 3 q^{9} - i q^{10} + 2 i q^{12} + 2 q^{14} + 2 i q^{15} + q^{16} - 3 i q^{18} + q^{20} - 4 q^{21} - 2 q^{24} + q^{25} + 4 i q^{27} + 2 i q^{28} - 2 q^{30} + i q^{32} + 2 i q^{35} + 3 q^{36} + i q^{40} - 4 i q^{42} + 3 q^{45} - 2 i q^{48} - 3 q^{49} + i q^{50} - 4 q^{54} - 2 q^{56} - 2 i q^{60} + 6 i q^{63} - q^{64} - i q^{67} - 2 q^{70} + 3 i q^{72} - 2 i q^{75} - q^{80} + 5 q^{81} + i q^{83} + 4 q^{84} + 3 i q^{90} + 2 q^{96} - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{9} + 4 q^{14} + 2 q^{16} + 2 q^{20} - 8 q^{21} - 4 q^{24} + 2 q^{25} - 4 q^{30} + 6 q^{36} + 6 q^{45} - 6 q^{49} - 8 q^{54} - 4 q^{56} - 2 q^{64} - 4 q^{70} - 2 q^{80} + 10 q^{81} + 8 q^{84} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2020\mathbb{Z}\right)^\times\).

\(n\) \(1011\) \(1617\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2019.1
1.00000i
1.00000i
1.00000i 2.00000i −1.00000 −1.00000 2.00000 2.00000i 1.00000i −3.00000 1.00000i
2019.2 1.00000i 2.00000i −1.00000 −1.00000 2.00000 2.00000i 1.00000i −3.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
404.d odd 2 1 CM by \(\Q(\sqrt{-101}) \)
505.d even 2 1 RM by \(\Q(\sqrt{505}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
101.b even 2 1 inner
2020.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2020.1.d.c 2
4.b odd 2 1 inner 2020.1.d.c 2
5.b even 2 1 inner 2020.1.d.c 2
20.d odd 2 1 CM 2020.1.d.c 2
101.b even 2 1 inner 2020.1.d.c 2
404.d odd 2 1 CM 2020.1.d.c 2
505.d even 2 1 RM 2020.1.d.c 2
2020.d odd 2 1 inner 2020.1.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2020.1.d.c 2 1.a even 1 1 trivial
2020.1.d.c 2 4.b odd 2 1 inner
2020.1.d.c 2 5.b even 2 1 inner
2020.1.d.c 2 20.d odd 2 1 CM
2020.1.d.c 2 101.b even 2 1 inner
2020.1.d.c 2 404.d odd 2 1 CM
2020.1.d.c 2 505.d even 2 1 RM
2020.1.d.c 2 2020.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2020, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{53} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
show more
show less