Properties

Label 4000.2.c.b
Level $4000$
Weight $2$
Character orbit 4000.c
Analytic conductor $31.940$
Analytic rank $0$
Dimension $8$
CM discriminant -20
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,2,Mod(1249,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.c (of order \(2\), degree \(1\), 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: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9} + (\beta_{6} + 3) q^{21} + ( - 2 \beta_{7} - \beta_{5} + \cdots + \beta_{2}) q^{23}+ \cdots + ( - 2 \beta_{6} - 4 \beta_{4} - 3 \beta_1) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 24 q^{21} - 12 q^{29} - 20 q^{41} + 4 q^{49} + 60 q^{61} - 48 q^{69} + 8 q^{81} + 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{20}^{6} + \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + 2\zeta_{20}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{5} - 2\zeta_{20}^{3} + 4\zeta_{20} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{3} + 2\zeta_{20}^{2} - 1 \) Copy content Toggle raw display
\(\zeta_{20}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.951057 + 0.309017i
0.587785 0.809017i
−0.951057 0.309017i
−0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 + 0.309017i
0.587785 + 0.809017i
0.951057 0.309017i
0 2.90211i 0 0 0 2.34458i 0 −5.42226 0
1249.2 0 2.17557i 0 0 0 2.45965i 0 −1.73311 0
1249.3 0 0.902113i 0 0 0 0.891491i 0 2.18619 0
1249.4 0 0.175571i 0 0 0 3.69572i 0 2.96917 0
1249.5 0 0.175571i 0 0 0 3.69572i 0 2.96917 0
1249.6 0 0.902113i 0 0 0 0.891491i 0 2.18619 0
1249.7 0 2.17557i 0 0 0 2.45965i 0 −1.73311 0
1249.8 0 2.90211i 0 0 0 2.34458i 0 −5.42226 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.c.b 8
4.b odd 2 1 inner 4000.2.c.b 8
5.b even 2 1 inner 4000.2.c.b 8
5.c odd 4 1 4000.2.a.b 4
5.c odd 4 1 4000.2.a.i yes 4
20.d odd 2 1 CM 4000.2.c.b 8
20.e even 4 1 4000.2.a.b 4
20.e even 4 1 4000.2.a.i yes 4
40.i odd 4 1 8000.2.a.z 4
40.i odd 4 1 8000.2.a.bs 4
40.k even 4 1 8000.2.a.z 4
40.k even 4 1 8000.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.b 4 5.c odd 4 1
4000.2.a.b 4 20.e even 4 1
4000.2.a.i yes 4 5.c odd 4 1
4000.2.a.i yes 4 20.e even 4 1
4000.2.c.b 8 1.a even 1 1 trivial
4000.2.c.b 8 4.b odd 2 1 inner
4000.2.c.b 8 5.b even 2 1 inner
4000.2.c.b 8 20.d odd 2 1 CM
8000.2.a.z 4 40.i odd 4 1
8000.2.a.z 4 40.k even 4 1
8000.2.a.bs 4 40.i odd 4 1
8000.2.a.bs 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\):

\( T_{3}^{8} + 14T_{3}^{6} + 51T_{3}^{4} + 34T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 26T_{7}^{6} + 211T_{7}^{4} + 606T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 26 T^{6} + \cdots + 361 \) 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
$23$ \( T^{8} + 154 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + \cdots + 281)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{3} + \cdots + 4705)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 254 T^{6} + \cdots + 160801 \) Copy content Toggle raw display
$47$ \( T^{8} + 166 T^{6} + \cdots + 1343281 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 30 T^{3} + \cdots - 3895)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 268 T^{2} + 13456)^{2} \) 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
$83$ \( T^{8} + \cdots + 3142611481 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + \cdots + 24881)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less