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Properties

Label 2020.1.ca.a
Level $2020$
Weight $1$
Character orbit 2020.ca
Analytic conductor $1.008$
Analytic rank $0$
Dimension $40$
Projective image $D_{100}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2020,1,Mod(3,2020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2020, base_ring=CyclotomicField(100))
 
chi = DirichletCharacter(H, H._module([50, 75, 69]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2020.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2020 = 2^{2} \cdot 5 \cdot 101 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2020.ca (of order \(100\), degree \(40\), 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: \(40\)
Coefficient field: \(\Q(\zeta_{100})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{40} - x^{30} + x^{20} - x^{10} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{100}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{100} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{100}^{22} q^{2} + \zeta_{100}^{44} q^{4} - \zeta_{100}^{49} q^{5} - \zeta_{100}^{16} q^{8} - \zeta_{100}^{36} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{100}^{22} q^{2} + \zeta_{100}^{44} q^{4} - \zeta_{100}^{49} q^{5} - \zeta_{100}^{16} q^{8} - \zeta_{100}^{36} q^{9} + \zeta_{100}^{21} q^{10} + ( - \zeta_{100}^{20} - \zeta_{100}^{9}) q^{13} - \zeta_{100}^{38} q^{16} + ( - \zeta_{100}^{40} + \zeta_{100}^{5}) q^{17} + \zeta_{100}^{8} q^{18} + \zeta_{100}^{43} q^{20} - \zeta_{100}^{48} q^{25} + ( - \zeta_{100}^{42} - \zeta_{100}^{31}) q^{26} + (\zeta_{100}^{17} - \zeta_{100}^{12}) q^{29} + \zeta_{100}^{10} q^{32} + (\zeta_{100}^{27} + \zeta_{100}^{12}) q^{34} + \zeta_{100}^{30} q^{36} + (\zeta_{100}^{38} - \zeta_{100}) q^{37} - \zeta_{100}^{15} q^{40} + ( - \zeta_{100}^{3} + \zeta_{100}^{2}) q^{41} - \zeta_{100}^{35} q^{45} - \zeta_{100}^{46} q^{49} + \zeta_{100}^{20} q^{50} + (\zeta_{100}^{14} + \zeta_{100}^{3}) q^{52} + (\zeta_{100}^{33} - \zeta_{100}^{29}) q^{53} + (\zeta_{100}^{39} - \zeta_{100}^{34}) q^{58} + (\zeta_{100}^{35} + \zeta_{100}^{28}) q^{61} + \zeta_{100}^{32} q^{64} + ( - \zeta_{100}^{19} - \zeta_{100}^{8}) q^{65} + (\zeta_{100}^{49} + \zeta_{100}^{34}) q^{68} - \zeta_{100}^{2} q^{72} + ( - \zeta_{100}^{43} + \zeta_{100}^{41}) q^{73} + ( - \zeta_{100}^{23} - \zeta_{100}^{10}) q^{74} - \zeta_{100}^{37} q^{80} - \zeta_{100}^{22} q^{81} + ( - \zeta_{100}^{25} + \zeta_{100}^{24}) q^{82} + ( - \zeta_{100}^{39} + \zeta_{100}^{4}) q^{85} + (\zeta_{100}^{26} + \zeta_{100}^{23}) q^{89} + \zeta_{100}^{7} q^{90} + (\zeta_{100}^{46} - \zeta_{100}^{17}) q^{97} + \zeta_{100}^{18} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 10 q^{13} + 10 q^{17} + 10 q^{32} + 10 q^{36} - 10 q^{50} - 10 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1011\) \(1617\) \(1921\)
\(\chi(n)\) \(-1\) \(\zeta_{100}^{25}\) \(\zeta_{100}^{19}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.998027 − 0.0627905i
−0.982287 − 0.187381i
−0.368125 + 0.929776i
0.248690 − 0.968583i
0.904827 − 0.425779i
−0.125333 − 0.992115i
−0.904827 − 0.425779i
0.982287 − 0.187381i
0.481754 − 0.876307i
0.844328 − 0.535827i
0.684547 + 0.728969i
−0.684547 − 0.728969i
−0.844328 + 0.535827i
−0.481754 + 0.876307i
−0.982287 + 0.187381i
0.904827 + 0.425779i
0.125333 + 0.992115i
−0.904827 + 0.425779i
−0.248690 + 0.968583i
0.368125 − 0.929776i
0.187381 + 0.982287i 0 −0.929776 + 0.368125i −0.998027 + 0.0627905i 0 0 −0.535827 − 0.844328i 0.637424 − 0.770513i −0.248690 − 0.968583i
27.1 −0.535827 − 0.844328i 0 −0.425779 + 0.904827i −0.982287 + 0.187381i 0 0 0.992115 − 0.125333i −0.876307 − 0.481754i 0.684547 + 0.728969i
83.1 0.425779 − 0.904827i 0 −0.637424 − 0.770513i −0.368125 − 0.929776i 0 0 −0.968583 + 0.248690i −0.535827 − 0.844328i −0.998027 − 0.0627905i
103.1 −0.728969 + 0.684547i 0 0.0627905 − 0.998027i 0.248690 + 0.968583i 0 0 0.637424 + 0.770513i 0.929776 − 0.368125i −0.844328 − 0.535827i
127.1 −0.968583 + 0.248690i 0 0.876307 − 0.481754i 0.904827 + 0.425779i 0 0 −0.728969 + 0.684547i 0.992115 − 0.125333i −0.982287 − 0.187381i
147.1 0.929776 + 0.368125i 0 0.728969 + 0.684547i −0.125333 + 0.992115i 0 0 0.425779 + 0.904827i 0.187381 − 0.982287i −0.481754 + 0.876307i
167.1 −0.968583 − 0.248690i 0 0.876307 + 0.481754i −0.904827 + 0.425779i 0 0 −0.728969 − 0.684547i 0.992115 + 0.125333i 0.982287 − 0.187381i
187.1 −0.535827 + 0.844328i 0 −0.425779 − 0.904827i 0.982287 + 0.187381i 0 0 0.992115 + 0.125333i −0.876307 + 0.481754i −0.684547 + 0.728969i
263.1 −0.0627905 + 0.998027i 0 −0.992115 − 0.125333i 0.481754 + 0.876307i 0 0 0.187381 − 0.982287i −0.728969 + 0.684547i −0.904827 + 0.425779i
463.1 0.992115 + 0.125333i 0 0.968583 + 0.248690i 0.844328 + 0.535827i 0 0 0.929776 + 0.368125i −0.0627905 + 0.998027i 0.770513 + 0.637424i
467.1 0.637424 − 0.770513i 0 −0.187381 − 0.982287i 0.684547 − 0.728969i 0 0 −0.876307 − 0.481754i 0.425779 + 0.904827i −0.125333 − 0.992115i
543.1 0.637424 − 0.770513i 0 −0.187381 − 0.982287i −0.684547 + 0.728969i 0 0 −0.876307 − 0.481754i 0.425779 + 0.904827i 0.125333 + 0.992115i
547.1 0.992115 + 0.125333i 0 0.968583 + 0.248690i −0.844328 − 0.535827i 0 0 0.929776 + 0.368125i −0.0627905 + 0.998027i −0.770513 − 0.637424i
747.1 −0.0627905 + 0.998027i 0 −0.992115 − 0.125333i −0.481754 − 0.876307i 0 0 0.187381 − 0.982287i −0.728969 + 0.684547i 0.904827 − 0.425779i
823.1 −0.535827 + 0.844328i 0 −0.425779 − 0.904827i −0.982287 − 0.187381i 0 0 0.992115 + 0.125333i −0.876307 + 0.481754i 0.684547 − 0.728969i
843.1 −0.968583 − 0.248690i 0 0.876307 + 0.481754i 0.904827 − 0.425779i 0 0 −0.728969 − 0.684547i 0.992115 + 0.125333i −0.982287 + 0.187381i
863.1 0.929776 + 0.368125i 0 0.728969 + 0.684547i 0.125333 − 0.992115i 0 0 0.425779 + 0.904827i 0.187381 − 0.982287i 0.481754 − 0.876307i
883.1 −0.968583 + 0.248690i 0 0.876307 − 0.481754i −0.904827 − 0.425779i 0 0 −0.728969 + 0.684547i 0.992115 − 0.125333i 0.982287 + 0.187381i
907.1 −0.728969 + 0.684547i 0 0.0627905 − 0.998027i −0.248690 − 0.968583i 0 0 0.637424 + 0.770513i 0.929776 − 0.368125i 0.844328 + 0.535827i
927.1 0.425779 − 0.904827i 0 −0.637424 − 0.770513i 0.368125 + 0.929776i 0 0 −0.968583 + 0.248690i −0.535827 − 0.844328i 0.998027 + 0.0627905i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.1
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.ba even 100 1 inner
2020.ca odd 100 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2020.1.ca.a ✓ 40
4.b odd 2 1 CM 2020.1.ca.a ✓ 40
5.c odd 4 1 2020.1.cd.a yes 40
20.e even 4 1 2020.1.cd.a yes 40
101.i odd 100 1 2020.1.cd.a yes 40
404.r even 100 1 2020.1.cd.a yes 40
505.ba even 100 1 inner 2020.1.ca.a ✓ 40
2020.ca odd 100 1 inner 2020.1.ca.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2020.1.ca.a ✓ 40 1.a even 1 1 trivial
2020.1.ca.a ✓ 40 4.b odd 2 1 CM
2020.1.ca.a ✓ 40 505.ba even 100 1 inner
2020.1.ca.a ✓ 40 2020.ca odd 100 1 inner
2020.1.cd.a yes 40 5.c odd 4 1
2020.1.cd.a yes 40 20.e even 4 1
2020.1.cd.a yes 40 101.i odd 100 1
2020.1.cd.a yes 40 404.r even 100 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{20} - T^{15} + T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{40} \) Copy content Toggle raw display
$5$ \( T^{40} - T^{30} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{40} \) Copy content Toggle raw display
$11$ \( T^{40} \) Copy content Toggle raw display
$13$ \( T^{40} - 10 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{40} \) Copy content Toggle raw display
$23$ \( T^{40} \) Copy content Toggle raw display
$29$ \( T^{40} + 8 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{40} \) Copy content Toggle raw display
$37$ \( T^{40} + 10 T^{37} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{40} - 15 T^{36} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{40} \) Copy content Toggle raw display
$47$ \( T^{40} \) Copy content Toggle raw display
$53$ \( T^{40} - 10 T^{36} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{40} \) Copy content Toggle raw display
$61$ \( T^{40} - 5 T^{38} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{40} \) Copy content Toggle raw display
$71$ \( T^{40} \) Copy content Toggle raw display
$73$ \( T^{40} + 10 T^{36} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{40} \) Copy content Toggle raw display
$83$ \( T^{40} \) Copy content Toggle raw display
$89$ \( T^{40} - 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{40} + 10 T^{37} + \cdots + 1 \) Copy content Toggle raw display
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