↑

Properties

Label 2500.1.d.a
Level $2500$
Weight $1$
Character orbit 2500.d
Analytic conductor $1.248$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,1,Mod(2499,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2500, 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("2500.2499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2500.d (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: \(1.24766253158\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6250000.1

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - q^{4} - \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - q^{4} - \beta_{3} q^{8} - q^{9} - \beta_1 q^{13} + q^{16} + ( - \beta_{3} - \beta_1) q^{17} - \beta_{3} q^{18} + \beta_{2} q^{26} + (\beta_{2} + 1) q^{29} + \beta_{3} q^{32} + (\beta_{2} + 1) q^{34} + q^{36} + ( - \beta_{3} - \beta_1) q^{37} + \beta_{2} q^{41} - q^{49} + \beta_1 q^{52} - \beta_1 q^{53} + (\beta_{3} + \beta_1) q^{58} + \beta_{2} q^{61} - q^{64} + (\beta_{3} + \beta_1) q^{68} + \beta_{3} q^{72} - \beta_1 q^{73} + (\beta_{2} + 1) q^{74} + q^{81} + \beta_1 q^{82} + (\beta_{2} + 1) q^{89} + ( - \beta_{3} - \beta_1) q^{97} - \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 2 q^{26} + 2 q^{29} + 2 q^{34} + 4 q^{36} - 2 q^{41} - 4 q^{49} - 2 q^{61} - 4 q^{64} + 2 q^{74} + 4 q^{81} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2499.1
1.61803i
− 0.618034i
0.618034i
− 1.61803i
− 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.2 − 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.3 1.00000i 0 −1.00000 0 0 0 − 1.00000i −1.00000 0
2499.4 1.00000i 0 −1.00000 0 0 0 − 1.00000i −1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.1.d.a 4
4.b odd 2 1 CM 2500.1.d.a 4
5.b even 2 1 inner 2500.1.d.a 4
5.c odd 4 1 2500.1.b.a 2
5.c odd 4 1 2500.1.b.b 2
20.d odd 2 1 inner 2500.1.d.a 4
20.e even 4 1 2500.1.b.a 2
20.e even 4 1 2500.1.b.b 2
25.d even 5 2 500.1.h.a 8
25.d even 5 2 2500.1.h.e 8
25.e even 10 2 500.1.h.a 8
25.e even 10 2 2500.1.h.e 8
25.f odd 20 2 100.1.j.a ✓ 4
25.f odd 20 2 500.1.j.a 4
25.f odd 20 2 2500.1.j.a 4
25.f odd 20 2 2500.1.j.b 4
75.l even 20 2 900.1.x.a 4
100.h odd 10 2 500.1.h.a 8
100.h odd 10 2 2500.1.h.e 8
100.j odd 10 2 500.1.h.a 8
100.j odd 10 2 2500.1.h.e 8
100.l even 20 2 100.1.j.a ✓ 4
100.l even 20 2 500.1.j.a 4
100.l even 20 2 2500.1.j.a 4
100.l even 20 2 2500.1.j.b 4
200.v even 20 2 1600.1.bh.a 4
200.x odd 20 2 1600.1.bh.a 4
300.u odd 20 2 900.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a ✓ 4 25.f odd 20 2
100.1.j.a ✓ 4 100.l even 20 2
500.1.h.a 8 25.d even 5 2
500.1.h.a 8 25.e even 10 2
500.1.h.a 8 100.h odd 10 2
500.1.h.a 8 100.j odd 10 2
500.1.j.a 4 25.f odd 20 2
500.1.j.a 4 100.l even 20 2
900.1.x.a 4 75.l even 20 2
900.1.x.a 4 300.u odd 20 2
1600.1.bh.a 4 200.v even 20 2
1600.1.bh.a 4 200.x odd 20 2
2500.1.b.a 2 5.c odd 4 1
2500.1.b.a 2 20.e even 4 1
2500.1.b.b 2 5.c odd 4 1
2500.1.b.b 2 20.e even 4 1
2500.1.d.a 4 1.a even 1 1 trivial
2500.1.d.a 4 4.b odd 2 1 CM
2500.1.d.a 4 5.b even 2 1 inner
2500.1.d.a 4 20.d odd 2 1 inner
2500.1.h.e 8 25.d even 5 2
2500.1.h.e 8 25.e even 10 2
2500.1.h.e 8 100.h odd 10 2
2500.1.h.e 8 100.j odd 10 2
2500.1.j.a 4 25.f odd 20 2
2500.1.j.a 4 100.l even 20 2
2500.1.j.b 4 25.f odd 20 2
2500.1.j.b 4 100.l even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2500, [\chi])\).

Hecke characteristic polynomials

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