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Properties

Label 2500.4.a.g
Level $2500$
Weight $4$
Character orbit 2500.a
Self dual yes
Analytic conductor $147.505$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,4,Mod(1,2500)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2500.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2500.a (trivial)

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: yes
Analytic conductor: \(147.504775014\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 100)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 328 q^{9} + 60 q^{11} + 128 q^{19} + 312 q^{21} + 848 q^{29} + 312 q^{31} + 628 q^{39} + 1352 q^{41} + 2796 q^{49} + 1664 q^{51} + 1616 q^{59} + 1204 q^{61} + 1644 q^{69} - 944 q^{71} + 184 q^{79} + 5048 q^{81} + 2392 q^{89} + 1588 q^{91} + 2540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −9.83716 0 0 0 −24.7439 0 69.7698 0
1.2 0 −9.79423 0 0 0 −0.760275 0 68.9269 0
1.3 0 −8.63719 0 0 0 24.1794 0 47.6010 0
1.4 0 −8.56164 0 0 0 −36.0715 0 46.3016 0
1.5 0 −7.24235 0 0 0 −4.97648 0 25.4516 0
1.6 0 −7.02723 0 0 0 21.6979 0 22.3820 0
1.7 0 −6.95286 0 0 0 −19.2903 0 21.3422 0
1.8 0 −5.27124 0 0 0 9.22290 0 0.785949 0
1.9 0 −4.95605 0 0 0 −16.8592 0 −2.43753 0
1.10 0 −4.53906 0 0 0 36.0979 0 −6.39697 0
1.11 0 −3.73656 0 0 0 30.1378 0 −13.0381 0
1.12 0 −2.93376 0 0 0 7.35306 0 −18.3931 0
1.13 0 −2.25419 0 0 0 −16.0147 0 −21.9186 0
1.14 0 −1.68733 0 0 0 −19.2807 0 −24.1529 0
1.15 0 −1.05300 0 0 0 15.9472 0 −25.8912 0
1.16 0 −0.816909 0 0 0 1.56595 0 −26.3327 0
1.17 0 0.816909 0 0 0 −1.56595 0 −26.3327 0
1.18 0 1.05300 0 0 0 −15.9472 0 −25.8912 0
1.19 0 1.68733 0 0 0 19.2807 0 −24.1529 0
1.20 0 2.25419 0 0 0 16.0147 0 −21.9186 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 2500.4.a.g 32
5.b even 2 1 inner 2500.4.a.g 32
25.d even 5 2 500.4.g.b 64
25.e even 10 2 500.4.g.b 64
25.f odd 20 2 100.4.i.a ✓ 32
25.f odd 20 2 500.4.i.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.i.a ✓ 32 25.f odd 20 2
500.4.g.b 64 25.d even 5 2
500.4.g.b 64 25.e even 10 2
500.4.i.a 32 25.f odd 20 2
2500.4.a.g 32 1.a even 1 1 trivial
2500.4.a.g 32 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 596 T_{3}^{30} + 158040 T_{3}^{28} - 24630320 T_{3}^{26} + 2511163320 T_{3}^{24} + \cdots + 11\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2500))\). Copy content Toggle raw display