Properties

Label 1027.1.x.a
Level $1027$
Weight $1$
Character orbit 1027.x
Analytic conductor $0.513$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1027,1,Mod(213,1027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1027, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1027.213");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1027 = 13 \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1027.x (of order \(12\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.13711477.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{3} - \zeta_{12}^{5} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{4} q^{3} - \zeta_{12}^{5} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{5} + \zeta_{12}^{3} q^{12} + \zeta_{12}^{4} q^{13} + (\zeta_{12}^{3} - 1) q^{15} - \zeta_{12}^{4} q^{16} - \zeta_{12}^{3} q^{17} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{19} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{20} + \zeta_{12} q^{25} - q^{27} - \zeta_{12}^{2} q^{29} - \zeta_{12}^{2} q^{39} - \zeta_{12} q^{43} + \zeta_{12}^{2} q^{48} + \zeta_{12} q^{49} + \zeta_{12} q^{51} + \zeta_{12}^{3} q^{52} + \zeta_{12}^{2} q^{53} + ( - \zeta_{12}^{3} + 1) q^{57} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{60} - q^{61} - \zeta_{12}^{3} q^{64} + (\zeta_{12}^{3} - 1) q^{65} + (\zeta_{12}^{3} - 1) q^{67} - \zeta_{12}^{2} q^{68} + ( - \zeta_{12}^{3} + 1) q^{71} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{73} + \zeta_{12}^{5} q^{75} + (\zeta_{12}^{4} - \zeta_{12}) q^{76} + \zeta_{12}^{2} q^{79} + ( - \zeta_{12}^{3} + 1) q^{80} - \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{83} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{85} + q^{87} + (\zeta_{12}^{4} - 2 \zeta_{12}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{13} - 4 q^{15} + 2 q^{16} - 2 q^{19} + 2 q^{20} - 4 q^{27} - 2 q^{29} - 2 q^{39} + 2 q^{48} + 2 q^{53} + 4 q^{57} + 2 q^{60} - 4 q^{61} - 4 q^{65} - 4 q^{67} - 2 q^{68} + 4 q^{71} + 2 q^{73} - 2 q^{76} + 2 q^{79} + 4 q^{80} + 2 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(80\) \(872\)
\(\chi(n)\) \(-\zeta_{12}^{3}\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
213.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.500000 + 0.866025i 0.866025 0.500000i 1.36603 + 0.366025i 0 0 0 0 0
450.1 0 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.366025 + 1.36603i 0 0 0 0 0
655.1 0 −0.500000 0.866025i −0.866025 0.500000i −0.366025 1.36603i 0 0 0 0 0
892.1 0 −0.500000 0.866025i 0.866025 + 0.500000i 1.36603 0.366025i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
79.c even 3 1 inner
1027.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1027.1.x.a 4
13.d odd 4 1 inner 1027.1.x.a 4
79.c even 3 1 inner 1027.1.x.a 4
1027.x odd 12 1 inner 1027.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1027.1.x.a 4 1.a even 1 1 trivial
1027.1.x.a 4 13.d odd 4 1 inner
1027.1.x.a 4 79.c even 3 1 inner
1027.1.x.a 4 1027.x odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1027, [\chi])\).

Hecke characteristic polynomials

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