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Properties

Label 500.3.d.e
Level $500$
Weight $3$
Character orbit 500.d
Analytic conductor $13.624$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,3,Mod(499,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.499");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 500.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: \(13.6240132180\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{4} - 18 q^{6} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{4} - 18 q^{6} + 144 q^{9} + 8 q^{14} + 18 q^{16} - 16 q^{21} + 38 q^{24} + 104 q^{26} - 48 q^{29} + 114 q^{34} - 40 q^{36} - 24 q^{41} - 80 q^{44} - 18 q^{46} + 264 q^{49} - 434 q^{54} + 62 q^{56} + 96 q^{61} - 504 q^{64} - 430 q^{66} + 176 q^{69} - 626 q^{74} - 630 q^{76} + 512 q^{81} - 624 q^{84} - 498 q^{86} + 72 q^{89} + 18 q^{94} + 32 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1 −1.99300 − 0.167153i 1.30398 3.94412 + 0.666273i 0 −2.59883 − 0.217964i 0.846062 −7.74927 − 1.98716i −7.29965 0
499.2 −1.99300 + 0.167153i 1.30398 3.94412 − 0.666273i 0 −2.59883 + 0.217964i 0.846062 −7.74927 + 1.98716i −7.29965 0
499.3 −1.90784 − 0.600134i 1.84848 3.27968 + 2.28991i 0 −3.52660 − 1.10934i 10.0454 −4.88283 − 6.33703i −5.58311 0
499.4 −1.90784 + 0.600134i 1.84848 3.27968 − 2.28991i 0 −3.52660 + 1.10934i 10.0454 −4.88283 + 6.33703i −5.58311 0
499.5 −1.87770 − 0.688660i −3.78208 3.05149 + 2.58619i 0 7.10161 + 2.60457i −1.63471 −3.94878 − 6.95753i 5.30414 0
499.6 −1.87770 + 0.688660i −3.78208 3.05149 − 2.58619i 0 7.10161 − 2.60457i −1.63471 −3.94878 + 6.95753i 5.30414 0
499.7 −1.81019 − 0.850426i −4.40247 2.55355 + 3.07886i 0 7.96928 + 3.74397i −5.32191 −2.00406 − 7.74492i 10.3817 0
499.8 −1.81019 + 0.850426i −4.40247 2.55355 − 3.07886i 0 7.96928 − 3.74397i −5.32191 −2.00406 + 7.74492i 10.3817 0
499.9 −1.79632 − 0.879336i 4.13523 2.45354 + 3.15914i 0 −7.42820 − 3.63626i −10.0590 −1.62939 − 7.83231i 8.10014 0
499.10 −1.79632 + 0.879336i 4.13523 2.45354 − 3.15914i 0 −7.42820 + 3.63626i −10.0590 −1.62939 + 7.83231i 8.10014 0
499.11 −1.59018 − 1.21298i 5.58157 1.05735 + 3.85772i 0 −8.87571 − 6.77035i −3.71470 2.99798 − 7.41702i 22.1540 0
499.12 −1.59018 + 1.21298i 5.58157 1.05735 − 3.85772i 0 −8.87571 + 6.77035i −3.71470 2.99798 + 7.41702i 22.1540 0
499.13 −1.41387 − 1.41456i −2.66132 −0.00195879 + 4.00000i 0 3.76275 + 3.76459i 13.7883 5.66101 − 5.65270i −1.91740 0
499.14 −1.41387 + 1.41456i −2.66132 −0.00195879 − 4.00000i 0 3.76275 − 3.76459i 13.7883 5.66101 + 5.65270i −1.91740 0
499.15 −1.16316 − 1.62698i −0.628132 −1.29412 + 3.78487i 0 0.730618 + 1.02196i −5.70922 7.66318 − 2.29689i −8.60545 0
499.16 −1.16316 + 1.62698i −0.628132 −1.29412 − 3.78487i 0 0.730618 − 1.02196i −5.70922 7.66318 + 2.29689i −8.60545 0
499.17 −0.902272 − 1.78491i 1.78379 −2.37181 + 3.22095i 0 −1.60946 − 3.18391i −2.13371 7.88912 + 1.32730i −5.81809 0
499.18 −0.902272 + 1.78491i 1.78379 −2.37181 − 3.22095i 0 −1.60946 + 3.18391i −2.13371 7.88912 − 1.32730i −5.81809 0
499.19 −0.487257 − 1.93974i −2.23496 −3.52516 + 1.89030i 0 1.08900 + 4.33523i 7.15930 5.38434 + 5.91683i −4.00498 0
499.20 −0.487257 + 1.93974i −2.23496 −3.52516 − 1.89030i 0 1.08900 − 4.33523i 7.15930 5.38434 − 5.91683i −4.00498 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
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 500.3.d.e 48
4.b odd 2 1 inner 500.3.d.e 48
5.b even 2 1 inner 500.3.d.e 48
5.c odd 4 1 500.3.b.c ✓ 24
5.c odd 4 1 500.3.b.e yes 24
20.d odd 2 1 inner 500.3.d.e 48
20.e even 4 1 500.3.b.c ✓ 24
20.e even 4 1 500.3.b.e yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.3.b.c ✓ 24 5.c odd 4 1
500.3.b.c ✓ 24 20.e even 4 1
500.3.b.e yes 24 5.c odd 4 1
500.3.b.e yes 24 20.e even 4 1
500.3.d.e 48 1.a even 1 1 trivial
500.3.d.e 48 4.b odd 2 1 inner
500.3.d.e 48 5.b even 2 1 inner
500.3.d.e 48 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 144 T_{3}^{22} + 8846 T_{3}^{20} - 304824 T_{3}^{18} + 6524471 T_{3}^{16} + \cdots + 11508998400 \) acting on \(S_{3}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display