↑

Properties

Label 1000.2.d.c
Level $1000$
Weight $2$
Character orbit 1000.d
Analytic conductor $7.985$
Analytic rank $0$
Dimension $40$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(501,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.501");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.d (of order \(2\), degree \(1\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{4} - 2 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{4} - 2 q^{6} - 24 q^{9} + 12 q^{14} + 18 q^{16} - 6 q^{24} + 20 q^{26} + 48 q^{31} - 6 q^{34} - 40 q^{36} + 8 q^{39} + 44 q^{41} + 8 q^{44} - 30 q^{46} + 12 q^{49} - 2 q^{54} + 50 q^{56} + 72 q^{64} + 42 q^{66} + 96 q^{71} + 6 q^{74} - 2 q^{76} + 96 q^{79} - 56 q^{81} + 116 q^{84} + 46 q^{86} - 44 q^{89} - 14 q^{94} + 12 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1 −1.41383 − 0.0329621i 3.08659i 1.99783 + 0.0932055i 0 0.101740 − 4.36391i −3.24242 −2.82151 − 0.197629i −6.52702 0
501.2 −1.41383 + 0.0329621i − 3.08659i 1.99783 − 0.0932055i 0 0.101740 + 4.36391i −3.24242 −2.82151 + 0.197629i −6.52702 0
501.3 −1.39686 − 0.220862i − 0.939252i 1.90244 + 0.617026i 0 −0.207445 + 1.31200i 1.90117 −2.52117 − 1.28208i 2.11781 0
501.4 −1.39686 + 0.220862i 0.939252i 1.90244 − 0.617026i 0 −0.207445 − 1.31200i 1.90117 −2.52117 + 1.28208i 2.11781 0
501.5 −1.36535 − 0.368522i 0.513027i 1.72838 + 1.00633i 0 0.189062 − 0.700464i 1.12889 −1.98900 − 2.01094i 2.73680 0
501.6 −1.36535 + 0.368522i − 0.513027i 1.72838 − 1.00633i 0 0.189062 + 0.700464i 1.12889 −1.98900 + 2.01094i 2.73680 0
501.7 −1.26751 − 0.627237i 1.69676i 1.21315 + 1.59005i 0 1.06427 − 2.15066i −4.40040 −0.540334 − 2.77634i 0.120998 0
501.8 −1.26751 + 0.627237i − 1.69676i 1.21315 − 1.59005i 0 1.06427 + 2.15066i −4.40040 −0.540334 + 2.77634i 0.120998 0
501.9 −1.23486 − 0.689292i − 2.52102i 1.04975 + 1.70236i 0 −1.73772 + 3.11311i 0.987050 −0.122873 − 2.82576i −3.35556 0
501.10 −1.23486 + 0.689292i 2.52102i 1.04975 − 1.70236i 0 −1.73772 − 3.11311i 0.987050 −0.122873 + 2.82576i −3.35556 0
501.11 −0.800179 − 1.16607i 2.62662i −0.719426 + 1.86613i 0 3.06282 − 2.10177i −0.269237 2.75170 − 0.654336i −3.89913 0
501.12 −0.800179 + 1.16607i − 2.62662i −0.719426 − 1.86613i 0 3.06282 + 2.10177i −0.269237 2.75170 + 0.654336i −3.89913 0
501.13 −0.739452 − 1.20549i − 2.07677i −0.906422 + 1.78281i 0 −2.50353 + 1.53567i 2.55636 2.81941 − 0.225614i −1.31298 0
501.14 −0.739452 + 1.20549i 2.07677i −0.906422 − 1.78281i 0 −2.50353 − 1.53567i 2.55636 2.81941 + 0.225614i −1.31298 0
501.15 −0.572751 − 1.29304i 0.207209i −1.34391 + 1.48118i 0 0.267929 − 0.118679i −2.73181 2.68496 + 0.889389i 2.95706 0
501.16 −0.572751 + 1.29304i − 0.207209i −1.34391 − 1.48118i 0 0.267929 + 0.118679i −2.73181 2.68496 − 0.889389i 2.95706 0
501.17 −0.493756 − 1.32522i − 1.83526i −1.51241 + 1.30867i 0 −2.43212 + 0.906170i 1.31564 2.48104 + 1.35811i −0.368171 0
501.18 −0.493756 + 1.32522i 1.83526i −1.51241 − 1.30867i 0 −2.43212 − 0.906170i 1.31564 2.48104 − 1.35811i −0.368171 0
501.19 −0.212863 − 1.39810i 1.21236i −1.90938 + 0.595207i 0 1.69500 − 0.258065i 4.63239 1.23860 + 2.54281i 1.53019 0
501.20 −0.212863 + 1.39810i − 1.21236i −1.90938 − 0.595207i 0 1.69500 + 0.258065i 4.63239 1.23860 − 2.54281i 1.53019 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.d.c ✓ 40
4.b odd 2 1 4000.2.d.c 40
5.b even 2 1 inner 1000.2.d.c ✓ 40
5.c odd 4 1 1000.2.f.c 20
5.c odd 4 1 1000.2.f.d 20
8.b even 2 1 inner 1000.2.d.c ✓ 40
8.d odd 2 1 4000.2.d.c 40
20.d odd 2 1 4000.2.d.c 40
20.e even 4 1 4000.2.f.c 20
20.e even 4 1 4000.2.f.d 20
40.e odd 2 1 4000.2.d.c 40
40.f even 2 1 inner 1000.2.d.c ✓ 40
40.i odd 4 1 1000.2.f.c 20
40.i odd 4 1 1000.2.f.d 20
40.k even 4 1 4000.2.f.c 20
40.k even 4 1 4000.2.f.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.d.c ✓ 40 1.a even 1 1 trivial
1000.2.d.c ✓ 40 5.b even 2 1 inner
1000.2.d.c ✓ 40 8.b even 2 1 inner
1000.2.d.c ✓ 40 40.f even 2 1 inner
1000.2.f.c 20 5.c odd 4 1
1000.2.f.c 20 40.i odd 4 1
1000.2.f.d 20 5.c odd 4 1
1000.2.f.d 20 40.i odd 4 1
4000.2.d.c 40 4.b odd 2 1
4000.2.d.c 40 8.d odd 2 1
4000.2.d.c 40 20.d odd 2 1
4000.2.d.c 40 40.e odd 2 1
4000.2.f.c 20 20.e even 4 1
4000.2.f.c 20 40.k even 4 1
4000.2.f.d 20 20.e even 4 1
4000.2.f.d 20 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 \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{20} + 36 T_{3}^{18} + 538 T_{3}^{16} + 4348 T_{3}^{14} + 20735 T_{3}^{12} + 59708 T_{3}^{10} + \cdots + 256 \) Copy content Toggle raw display
\( T_{7}^{20} - 73 T_{7}^{18} + 2133 T_{7}^{16} - 32388 T_{7}^{14} + 279482 T_{7}^{12} - 1409126 T_{7}^{10} + \cdots + 119961 \) Copy content Toggle raw display