Properties

Label 4000.1.p.a
Level $4000$
Weight $1$
Character orbit 4000.p
Analytic conductor $1.996$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,1,Mod(193,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.193");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.p (of order \(4\), 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: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of 20.2.400000000000000000000000000.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + ( - \zeta_{20}^{8} + \zeta_{20}^{7}) q^{7} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + ( - \zeta_{20}^{8} + \zeta_{20}^{7}) q^{7} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9} + (\zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{2} - \zeta_{20}) q^{21} + (\zeta_{20}^{8} + \zeta_{20}^{7}) q^{23} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{27} + (\zeta_{20}^{9} + \zeta_{20}) q^{29} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{41} + (\zeta_{20}^{9} + \zeta_{20}^{6}) q^{43} + (\zeta_{20}^{4} + \zeta_{20}) q^{47} + ( - \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4}) q^{49} + (\zeta_{20}^{7} - \zeta_{20}^{3}) q^{61} + (\zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}^{2} + 1) q^{63} + ( - \zeta_{20}^{5} - 1) q^{67} + ( - \zeta_{20}^{9} - \zeta_{20}^{8} - \zeta_{20}^{2} - \zeta_{20}) q^{69} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{3} + 1) q^{81} + ( - \zeta_{20}^{3} + \zeta_{20}^{2}) q^{83} + (\zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}^{2} + 1) q^{87} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{7} + 4 q^{21} - 2 q^{23} + 2 q^{43} - 2 q^{47} + 12 q^{63} - 8 q^{67} - 12 q^{81} + 2 q^{83} + 6 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.587785 + 0.809017i
0.951057 0.309017i
−0.587785 + 0.809017i
−0.951057 0.309017i
0.587785 0.809017i
0.951057 + 0.309017i
−0.587785 0.809017i
−0.951057 + 0.309017i
0 −1.39680 1.39680i 0 0 0 0.642040 0.642040i 0 2.90211i 0
193.2 0 −0.642040 0.642040i 0 0 0 0.221232 0.221232i 0 0.175571i 0
193.3 0 −0.221232 0.221232i 0 0 0 −1.26007 + 1.26007i 0 0.902113i 0
193.4 0 1.26007 + 1.26007i 0 0 0 1.39680 1.39680i 0 2.17557i 0
1057.1 0 −1.39680 + 1.39680i 0 0 0 0.642040 + 0.642040i 0 2.90211i 0
1057.2 0 −0.642040 + 0.642040i 0 0 0 0.221232 + 0.221232i 0 0.175571i 0
1057.3 0 −0.221232 + 0.221232i 0 0 0 −1.26007 1.26007i 0 0.902113i 0
1057.4 0 1.26007 1.26007i 0 0 0 1.39680 + 1.39680i 0 2.17557i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.p.a 8
4.b odd 2 1 4000.1.p.b yes 8
5.b even 2 1 4000.1.p.b yes 8
5.c odd 4 1 inner 4000.1.p.a 8
5.c odd 4 1 4000.1.p.b yes 8
20.d odd 2 1 CM 4000.1.p.a 8
20.e even 4 1 inner 4000.1.p.a 8
20.e even 4 1 4000.1.p.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.1.p.a 8 1.a even 1 1 trivial
4000.1.p.a 8 5.c odd 4 1 inner
4000.1.p.a 8 20.d odd 2 1 CM
4000.1.p.a 8 20.e even 4 1 inner
4000.1.p.b yes 8 4.b odd 2 1
4000.1.p.b yes 8 5.b even 2 1
4000.1.p.b yes 8 5.c odd 4 1
4000.1.p.b yes 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 11T_{3}^{4} + 20T_{3}^{3} + 18T_{3}^{2} + 6T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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