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Properties

Label 2020.1.d.e
Level $2020$
Weight $1$
Character orbit 2020.d
Analytic conductor $1.008$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -404
Inner twists $8$

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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: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{28}^{7} q^{2} + ( - \zeta_{28}^{11} - \zeta_{28}^{3}) q^{3} - q^{4} + \zeta_{28}^{2} q^{5} + ( - \zeta_{28}^{10} + \zeta_{28}^{4}) q^{6} + (\zeta_{28}^{13} + \zeta_{28}) q^{7} - \zeta_{28}^{7} q^{8} + ( - \zeta_{28}^{8} + \zeta_{28}^{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{28}^{7} q^{2} + ( - \zeta_{28}^{11} - \zeta_{28}^{3}) q^{3} - q^{4} + \zeta_{28}^{2} q^{5} + ( - \zeta_{28}^{10} + \zeta_{28}^{4}) q^{6} + (\zeta_{28}^{13} + \zeta_{28}) q^{7} - \zeta_{28}^{7} q^{8} + ( - \zeta_{28}^{8} + \zeta_{28}^{6} - 1) q^{9} + \zeta_{28}^{9} q^{10} + (\zeta_{28}^{9} - \zeta_{28}^{5}) q^{11} + (\zeta_{28}^{11} + \zeta_{28}^{3}) q^{12} + ( - \zeta_{28}^{8} - \zeta_{28}^{6}) q^{13} + (\zeta_{28}^{8} - \zeta_{28}^{6}) q^{14} + ( - \zeta_{28}^{13} - \zeta_{28}^{5}) q^{15} + q^{16} + ( - \zeta_{28}^{10} - \zeta_{28}^{4}) q^{17} + (\zeta_{28}^{13} - \zeta_{28}^{7} + \zeta_{28}) q^{18} - \zeta_{28}^{2} q^{20} + ( - \zeta_{28}^{12} + \cdots + \zeta_{28}^{2}) q^{21} + \cdots + (\zeta_{28}^{13} - \zeta_{28}^{11} + \cdots - \zeta_{28}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} + 2 q^{5} - 4 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 2 q^{5} - 4 q^{6} - 8 q^{9} - 4 q^{14} + 12 q^{16} - 2 q^{20} + 8 q^{21} + 4 q^{24} - 2 q^{25} + 4 q^{30} + 8 q^{36} - 6 q^{45} - 8 q^{49} + 8 q^{54} + 4 q^{56} - 12 q^{64} + 4 q^{70} + 2 q^{80} + 4 q^{81} - 8 q^{84} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
−0.781831 + 0.623490i
0.781831 + 0.623490i
0.433884 − 0.900969i
−0.433884 − 0.900969i
0.974928 − 0.222521i
−0.974928 − 0.222521i
−0.974928 + 0.222521i
0.974928 + 0.222521i
−0.433884 + 0.900969i
0.433884 + 0.900969i
0.781831 − 0.623490i
−0.781831 − 0.623490i
− 1.00000i − 1.80194i −1.00000 0.222521 − 0.974928i −1.80194 1.24698i 1.00000i −2.24698 −0.974928 − 0.222521i
2019.2 − 1.00000i − 1.80194i −1.00000 0.222521 + 0.974928i −1.80194 1.24698i 1.00000i −2.24698 0.974928 − 0.222521i
2019.3 − 1.00000i − 0.445042i −1.00000 −0.623490 − 0.781831i −0.445042 − 1.80194i 1.00000i 0.801938 −0.781831 + 0.623490i
2019.4 − 1.00000i − 0.445042i −1.00000 −0.623490 + 0.781831i −0.445042 − 1.80194i 1.00000i 0.801938 0.781831 + 0.623490i
2019.5 − 1.00000i 1.24698i −1.00000 0.900969 − 0.433884i 1.24698 − 0.445042i 1.00000i −0.554958 −0.433884 − 0.900969i
2019.6 − 1.00000i 1.24698i −1.00000 0.900969 + 0.433884i 1.24698 − 0.445042i 1.00000i −0.554958 0.433884 − 0.900969i
2019.7 1.00000i − 1.24698i −1.00000 0.900969 − 0.433884i 1.24698 0.445042i − 1.00000i −0.554958 0.433884 + 0.900969i
2019.8 1.00000i − 1.24698i −1.00000 0.900969 + 0.433884i 1.24698 0.445042i − 1.00000i −0.554958 −0.433884 + 0.900969i
2019.9 1.00000i 0.445042i −1.00000 −0.623490 − 0.781831i −0.445042 1.80194i − 1.00000i 0.801938 0.781831 − 0.623490i
2019.10 1.00000i 0.445042i −1.00000 −0.623490 + 0.781831i −0.445042 1.80194i − 1.00000i 0.801938 −0.781831 − 0.623490i
2019.11 1.00000i 1.80194i −1.00000 0.222521 − 0.974928i −1.80194 − 1.24698i − 1.00000i −2.24698 0.974928 + 0.222521i
2019.12 1.00000i 1.80194i −1.00000 0.222521 + 0.974928i −1.80194 − 1.24698i − 1.00000i −2.24698 −0.974928 + 0.222521i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2019.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
404.d odd 2 1 CM by \(\Q(\sqrt{-101}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
101.b even 2 1 inner
505.d 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.e ✓ 12
4.b odd 2 1 inner 2020.1.d.e ✓ 12
5.b even 2 1 inner 2020.1.d.e ✓ 12
20.d odd 2 1 inner 2020.1.d.e ✓ 12
101.b even 2 1 inner 2020.1.d.e ✓ 12
404.d odd 2 1 CM 2020.1.d.e ✓ 12
505.d even 2 1 inner 2020.1.d.e ✓ 12
2020.d odd 2 1 inner 2020.1.d.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2020.1.d.e ✓ 12 1.a even 1 1 trivial
2020.1.d.e ✓ 12 4.b odd 2 1 inner
2020.1.d.e ✓ 12 5.b even 2 1 inner
2020.1.d.e ✓ 12 20.d odd 2 1 inner
2020.1.d.e ✓ 12 101.b even 2 1 inner
2020.1.d.e ✓ 12 404.d odd 2 1 CM
2020.1.d.e ✓ 12 505.d even 2 1 inner
2020.1.d.e ✓ 12 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 \(S_{1}^{\mathrm{new}}(2020, [\chi])\):

\( T_{3}^{6} + 5T_{3}^{4} + 6T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{53} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
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