↑

Properties

Label 1027.1.s.a
Level $1027$
Weight $1$
Character orbit 1027.s
Analytic conductor $0.513$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -79
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1027,1,Mod(394,1027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1027, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1027.394");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1027 = 13 \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1027.s (of order \(6\), 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: \(0.512539767974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.183061929427.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6}^{2} + 1) q^{2} + ( - \zeta_{6}^{2} - \zeta_{6} + 1) q^{4} + (\zeta_{6}^{2} + \zeta_{6}) q^{5} + ( - \zeta_{6}^{2} + \zeta_{6}) q^{8} + \zeta_{6}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6}^{2} + 1) q^{2} + ( - \zeta_{6}^{2} - \zeta_{6} + 1) q^{4} + (\zeta_{6}^{2} + \zeta_{6}) q^{5} + ( - \zeta_{6}^{2} + \zeta_{6}) q^{8} + \zeta_{6}^{2} q^{9} + (\zeta_{6}^{2} + \zeta_{6} + 1) q^{10} + ( - \zeta_{6}^{2} + 1) q^{11} - q^{13} + (\zeta_{6} + 1) q^{16} + (\zeta_{6}^{2} + \zeta_{6}) q^{18} + ( - \zeta_{6} - 1) q^{19} + (\zeta_{6} + 2) q^{20} + ( - 2 \zeta_{6}^{2} - \zeta_{6} + 1) q^{22} - \zeta_{6} q^{23} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{25} + (\zeta_{6}^{2} - 1) q^{26} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{31} + (\zeta_{6} + 1) q^{32} + (\zeta_{6}^{2} + \zeta_{6} + 1) q^{36} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{38} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{40} + ( - 2 \zeta_{6}^{2} - \zeta_{6} - 1) q^{44} + ( - \zeta_{6} - 1) q^{45} + ( - \zeta_{6} - 1) q^{46} + \zeta_{6} q^{49} + (2 \zeta_{6}^{2} + \zeta_{6} - 1) q^{50} + (\zeta_{6}^{2} + \zeta_{6} - 1) q^{52} + (\zeta_{6}^{2} + \zeta_{6} + 1) q^{55} + ( - \zeta_{6}^{2} - \zeta_{6} - 1) q^{62} + q^{64} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{65} + (\zeta_{6} + 1) q^{72} + (\zeta_{6}^{2} + \zeta_{6}) q^{73} + (2 \zeta_{6}^{2} - 1) q^{76} - q^{79} + ( - \zeta_{6}^{2} + 1) q^{80} - \zeta_{6} q^{81} + (\zeta_{6}^{2} + \zeta_{6}) q^{83} + ( - \zeta_{6}^{2} - \zeta_{6} - 1) q^{88} + (\zeta_{6}^{2} - \zeta_{6} - 2) q^{90} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{92} + ( - 2 \zeta_{6}^{2} - \zeta_{6} + 1) q^{95} + (\zeta_{6} + 1) q^{97} + (\zeta_{6} + 1) q^{98} + (\zeta_{6}^{2} + \zeta_{6}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 2 q^{4} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 2 q^{4} - q^{9} + 3 q^{10} + 3 q^{11} - 2 q^{13} - q^{16} - 3 q^{19} + 6 q^{20} + 3 q^{22} - q^{23} - 4 q^{25} - 3 q^{26} + 2 q^{36} - 6 q^{38} + 6 q^{40} - 3 q^{45} - 3 q^{46} + q^{49} - 6 q^{50} - 2 q^{52} + 3 q^{55} - 3 q^{62} + 2 q^{64} + 3 q^{72} - 6 q^{76} - 2 q^{79} + 3 q^{80} - q^{81} - 3 q^{88} - 6 q^{90} - 4 q^{92} + 3 q^{95} + 3 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1027\mathbb{Z}\right)^\times\).

\(n\) \(80\) \(872\)
\(\chi(n)\) \(-\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
394.1
0.500000 − 0.866025i
0.500000 + 0.866025i
1.50000 + 0.866025i 0 1.00000 + 1.73205i − 1.73205i 0 0 1.73205i −0.500000 − 0.866025i 1.50000 − 2.59808i
868.1 1.50000 − 0.866025i 0 1.00000 − 1.73205i 1.73205i 0 0 − 1.73205i −0.500000 + 0.866025i 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)
13.e even 6 1 inner
1027.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1027.1.s.a ✓ 2
13.e even 6 1 inner 1027.1.s.a ✓ 2
79.b odd 2 1 CM 1027.1.s.a ✓ 2
1027.s odd 6 1 inner 1027.1.s.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1027.1.s.a ✓ 2 1.a even 1 1 trivial
1027.1.s.a ✓ 2 13.e even 6 1 inner
1027.1.s.a ✓ 2 79.b odd 2 1 CM
1027.1.s.a ✓ 2 1027.s odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1027, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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