Properties

Label 2020.1.t.a
Level $2020$
Weight $1$
Character orbit 2020.t
Analytic conductor $1.008$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2020,1,Mod(1323,2020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2020, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2020.1323");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2020 = 2^{2} \cdot 5 \cdot 101 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2020.t (of order \(4\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.515150500.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - i q^{5} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - i q^{5} - q^{8} - q^{9} + i q^{10} + ( - i - 1) q^{13} + q^{16} + (i - 1) q^{17} + q^{18} - i q^{20} - q^{25} + (i + 1) q^{26} + (i - 1) q^{29} - q^{32} + ( - i + 1) q^{34} - q^{36} + ( - i - 1) q^{37} + i q^{40} + (i - 1) q^{41} + i q^{45} + q^{49} + q^{50} + ( - i - 1) q^{52} + ( - i + 1) q^{58} + ( - i + 1) q^{61} + q^{64} + (i - 1) q^{65} + (i - 1) q^{68} + q^{72} + i q^{73} + (i + 1) q^{74} - i q^{80} + q^{81} + ( - i + 1) q^{82} + (i + 1) q^{85} + ( - i - 1) q^{89} - i q^{90} + ( - i - 1) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{25} + 2 q^{26} - 2 q^{29} - 2 q^{32} + 2 q^{34} - 2 q^{36} - 2 q^{37} - 2 q^{41} + 2 q^{49} + 2 q^{50} - 2 q^{52} + 2 q^{58} + 2 q^{61} + 2 q^{64} - 2 q^{65} - 2 q^{68} + 2 q^{72} + 2 q^{74} + 2 q^{81} + 2 q^{82} + 2 q^{85} - 2 q^{89} - 2 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1323.1
1.00000i
1.00000i
−1.00000 0 1.00000 1.00000i 0 0 −1.00000 −1.00000 1.00000i
1707.1 −1.00000 0 1.00000 1.00000i 0 0 −1.00000 −1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
505.j even 4 1 inner
2020.t odd 4 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2020, [\chi])\).

Hecke characteristic polynomials

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