Properties

Label 2000.1.z.a
Level $2000$
Weight $1$
Character orbit 2000.z
Analytic conductor $0.998$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -4
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2000,1,Mod(351,2000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2000, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2000.351");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2000.z (of order \(10\), degree \(4\), not 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: \(0.998130025266\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 400)
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.195312500000000.4

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{20}^{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20}^{8} q^{9} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{13} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{17} + (\zeta_{20}^{6} + \zeta_{20}^{2}) q^{29} + (\zeta_{20}^{5} - \zeta_{20}) q^{37} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{41} + q^{49} + ( - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{53} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{61} + (\zeta_{20}^{3} + \zeta_{20}) q^{73} - \zeta_{20}^{6} q^{81} + ( - \zeta_{20}^{4} - 1) q^{89} + (\zeta_{20}^{7} + \zeta_{20}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{9} + 4 q^{29} + 4 q^{41} + 8 q^{49} + 4 q^{61} - 2 q^{81} - 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times\).

\(n\) \(501\) \(751\) \(1377\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{20}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.587785 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0 0 0 0 0 0 0 0.309017 + 0.951057i 0
351.2 0 0 0 0 0 0 0 0.309017 + 0.951057i 0
1151.1 0 0 0 0 0 0 0 0.309017 0.951057i 0
1151.2 0 0 0 0 0 0 0 0.309017 0.951057i 0
1551.1 0 0 0 0 0 0 0 −0.809017 0.587785i 0
1551.2 0 0 0 0 0 0 0 −0.809017 0.587785i 0
1951.1 0 0 0 0 0 0 0 −0.809017 + 0.587785i 0
1951.2 0 0 0 0 0 0 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2000.1.z.a 8
4.b odd 2 1 CM 2000.1.z.a 8
5.b even 2 1 inner 2000.1.z.a 8
5.c odd 4 1 400.1.x.a 4
5.c odd 4 1 2000.1.x.a 4
15.e even 4 1 3600.1.ct.a 4
20.d odd 2 1 inner 2000.1.z.a 8
20.e even 4 1 400.1.x.a 4
20.e even 4 1 2000.1.x.a 4
25.d even 5 1 inner 2000.1.z.a 8
25.e even 10 1 inner 2000.1.z.a 8
25.f odd 20 1 400.1.x.a 4
25.f odd 20 1 2000.1.x.a 4
40.i odd 4 1 1600.1.bf.a 4
40.k even 4 1 1600.1.bf.a 4
60.l odd 4 1 3600.1.ct.a 4
75.l even 20 1 3600.1.ct.a 4
100.h odd 10 1 inner 2000.1.z.a 8
100.j odd 10 1 inner 2000.1.z.a 8
100.l even 20 1 400.1.x.a 4
100.l even 20 1 2000.1.x.a 4
200.v even 20 1 1600.1.bf.a 4
200.x odd 20 1 1600.1.bf.a 4
300.u odd 20 1 3600.1.ct.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 5.c odd 4 1
400.1.x.a 4 20.e even 4 1
400.1.x.a 4 25.f odd 20 1
400.1.x.a 4 100.l even 20 1
1600.1.bf.a 4 40.i odd 4 1
1600.1.bf.a 4 40.k even 4 1
1600.1.bf.a 4 200.v even 20 1
1600.1.bf.a 4 200.x odd 20 1
2000.1.x.a 4 5.c odd 4 1
2000.1.x.a 4 20.e even 4 1
2000.1.x.a 4 25.f odd 20 1
2000.1.x.a 4 100.l even 20 1
2000.1.z.a 8 1.a even 1 1 trivial
2000.1.z.a 8 4.b odd 2 1 CM
2000.1.z.a 8 5.b even 2 1 inner
2000.1.z.a 8 20.d odd 2 1 inner
2000.1.z.a 8 25.d even 5 1 inner
2000.1.z.a 8 25.e even 10 1 inner
2000.1.z.a 8 100.h odd 10 1 inner
2000.1.z.a 8 100.j odd 10 1 inner
3600.1.ct.a 4 15.e even 4 1
3600.1.ct.a 4 60.l odd 4 1
3600.1.ct.a 4 75.l even 20 1
3600.1.ct.a 4 300.u odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2000, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
show more
show less