Properties

Label 1003.1.h.b
Level $1003$
Weight $1$
Character orbit 1003.h
Analytic conductor $0.501$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -59
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1003,1,Mod(117,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1003.117");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1003.h (of order \(8\), 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: \(0.500562207671\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{2} - \zeta_{24}) q^{3} - \zeta_{24}^{6} q^{4} + (\zeta_{24}^{10} + \zeta_{24}^{5}) q^{5} + (\zeta_{24}^{11} - \zeta_{24}^{10}) q^{7} + (\zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{2} - \zeta_{24}) q^{3} - \zeta_{24}^{6} q^{4} + (\zeta_{24}^{10} + \zeta_{24}^{5}) q^{5} + (\zeta_{24}^{11} - \zeta_{24}^{10}) q^{7} + (\zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{8} + \zeta_{24}^{7}) q^{12} + ( - \zeta_{24}^{11} - \zeta_{24}^{7} - \zeta_{24}^{6} + 1) q^{15} - q^{16} + \zeta_{24}^{3} q^{17} - \zeta_{24}^{9} q^{19} + ( - \zeta_{24}^{11} + \zeta_{24}^{4}) q^{20} + (\zeta_{24}^{11} + \zeta_{24}) q^{21} + (\zeta_{24}^{10} - \zeta_{24}^{8} - \zeta_{24}^{3}) q^{25} + ( - \zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3}) q^{27} + (\zeta_{24}^{5} - \zeta_{24}^{4}) q^{28} + ( - \zeta_{24}^{8} - \zeta_{24}^{7}) q^{29} + ( - \zeta_{24}^{9} + \zeta_{24}^{8} - \zeta_{24}^{4} + \zeta_{24}^{3}) q^{35} + ( - \zeta_{24}^{10} - \zeta_{24}^{9} - \zeta_{24}^{8}) q^{36} + (\zeta_{24}^{7} + \zeta_{24}^{2}) q^{41} + (\zeta_{24}^{9} + \zeta_{24}^{8} + \zeta_{24}^{7} - \zeta_{24}^{2} - \zeta_{24} - 1) q^{45} + (\zeta_{24}^{2} + \zeta_{24}) q^{48} + ( - \zeta_{24}^{10} + \zeta_{24}^{9} - \zeta_{24}^{8}) q^{49} + ( - \zeta_{24}^{5} - \zeta_{24}^{4}) q^{51} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{53} + (\zeta_{24}^{11} + \zeta_{24}^{10}) q^{57} + \zeta_{24}^{3} q^{59} + ( - \zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24} - 1) q^{60} + ( - \zeta_{24}^{3} + 1) q^{63} + \zeta_{24}^{6} q^{64} - \zeta_{24}^{9} q^{68} + ( - \zeta_{24}^{9} + \zeta_{24}^{6}) q^{71} + ( - \zeta_{24}^{11} + \zeta_{24}^{10} + \zeta_{24}^{9} + \zeta_{24}^{5} + \zeta_{24}^{4} + 1) q^{75} - \zeta_{24}^{3} q^{76} + ( - \zeta_{24}^{7} + \zeta_{24}^{2}) q^{79} + ( - \zeta_{24}^{10} - \zeta_{24}^{5}) q^{80} + (\zeta_{24}^{8} + \zeta_{24}^{7} + \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{4}) q^{81} + ( - \zeta_{24}^{7} + \zeta_{24}^{5}) q^{84} + (\zeta_{24}^{8} - \zeta_{24}) q^{85} + (\zeta_{24}^{10} + 2 \zeta_{24}^{9} + \zeta_{24}^{8}) q^{87} + (\zeta_{24}^{7} + \zeta_{24}^{2}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 4 q^{12} + 8 q^{15} - 8 q^{16} + 4 q^{20} + 4 q^{25} - 4 q^{27} - 4 q^{28} + 4 q^{29} - 8 q^{35} + 4 q^{36} - 12 q^{45} + 4 q^{49} - 4 q^{51} - 8 q^{60} + 8 q^{63} + 12 q^{75} - 4 q^{85} - 4 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(120\) \(768\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
0 0.0999004 + 0.241181i 1.00000i −1.12484 + 0.465926i 0 1.83195 + 0.758819i 0 0.658919 0.658919i 0
117.2 0 0.607206 + 1.46593i 1.00000i 1.83195 0.758819i 0 −1.12484 0.465926i 0 −1.07313 + 1.07313i 0
648.1 0 −1.83195 + 0.758819i 1.00000i −0.607206 1.46593i 0 −0.0999004 + 0.241181i 0 2.07313 2.07313i 0
648.2 0 1.12484 0.465926i 1.00000i −0.0999004 0.241181i 0 −0.607206 + 1.46593i 0 0.341081 0.341081i 0
825.1 0 −1.83195 0.758819i 1.00000i −0.607206 + 1.46593i 0 −0.0999004 0.241181i 0 2.07313 + 2.07313i 0
825.2 0 1.12484 + 0.465926i 1.00000i −0.0999004 + 0.241181i 0 −0.607206 1.46593i 0 0.341081 + 0.341081i 0
943.1 0 0.0999004 0.241181i 1.00000i −1.12484 0.465926i 0 1.83195 0.758819i 0 0.658919 + 0.658919i 0
943.2 0 0.607206 1.46593i 1.00000i 1.83195 + 0.758819i 0 −1.12484 + 0.465926i 0 −1.07313 1.07313i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.2
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.d even 8 1 inner
1003.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1003.1.h.b 8
17.d even 8 1 inner 1003.1.h.b 8
59.b odd 2 1 CM 1003.1.h.b 8
1003.h odd 8 1 inner 1003.1.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.1.h.b 8 1.a even 1 1 trivial
1003.1.h.b 8 17.d even 8 1 inner
1003.1.h.b 8 59.b odd 2 1 CM
1003.1.h.b 8 1003.h odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{6} + 4T_{3}^{5} + 2T_{3}^{4} - 16T_{3}^{3} + 18T_{3}^{2} - 4T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1003, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{6} - 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{6} - 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + 10 T^{6} - 16 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} - 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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