Properties

Label 1003.1.b.b
Level $1003$
Weight $1$
Character orbit 1003.b
Analytic conductor $0.501$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -59
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1003,1,Mod(1002,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1003.1002");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1003.b (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: \(0.500562207671\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.17102153.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6}^{2} + \zeta_{6}) q^{3} + q^{4} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{5} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6}^{2} + \zeta_{6}) q^{3} + q^{4} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{5} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{9} + (\zeta_{6}^{2} + \zeta_{6}) q^{12} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{15} + q^{16} - q^{17} + q^{19} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{20} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{21} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{25} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{27} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{28} + (\zeta_{6}^{2} + \zeta_{6}) q^{29} + (\zeta_{6}^{2} - \zeta_{6} - 2) q^{35} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{36} + (\zeta_{6}^{2} + \zeta_{6}) q^{41} + (2 \zeta_{6}^{2} + \zeta_{6}) q^{45} + (\zeta_{6}^{2} + \zeta_{6}) q^{48} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{51} + q^{53} + (\zeta_{6}^{2} + \zeta_{6}) q^{57} - q^{59} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{60} + (2 \zeta_{6}^{2} + \zeta_{6}) q^{63} + q^{64} - q^{68} + ( - 2 \zeta_{6}^{2} - \zeta_{6}) q^{75} + q^{76} + (\zeta_{6}^{2} + \zeta_{6}) q^{79} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{80} + q^{81} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{84} + (\zeta_{6}^{2} + \zeta_{6}) q^{85} + (\zeta_{6}^{2} - \zeta_{6} - 2) q^{87} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{9} + 6 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 6 q^{21} - 4 q^{25} - 6 q^{35} - 4 q^{36} - 4 q^{49} + 2 q^{53} - 2 q^{59} + 6 q^{60} + 2 q^{64} - 2 q^{68} + 2 q^{76} + 2 q^{81} + 6 q^{84} - 6 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1003\mathbb{Z}\right)^\times\).

\(n\) \(120\) \(768\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1002.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 1.00000 1.73205i 0 1.73205i 0 −2.00000 0
1002.2 0 1.73205i 1.00000 1.73205i 0 1.73205i 0 −2.00000 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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
17.b even 2 1 inner
1003.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1003.1.b.b 2
17.b even 2 1 inner 1003.1.b.b 2
59.b odd 2 1 CM 1003.1.b.b 2
1003.b odd 2 1 inner 1003.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.1.b.b 2 1.a even 1 1 trivial
1003.1.b.b 2 17.b even 2 1 inner
1003.1.b.b 2 59.b odd 2 1 CM
1003.1.b.b 2 1003.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1003, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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