Properties

Label 1000.2.k.f
Level $1000$
Weight $2$
Character orbit 1000.k
Analytic conductor $7.985$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(307,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.k (of order \(4\), degree \(2\), 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: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 56 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 56 q^{6} + 24 q^{11} + 24 q^{16} - 60 q^{26} - 152 q^{36} + 24 q^{41} - 20 q^{46} + 72 q^{51} - 20 q^{56} + 72 q^{66} + 124 q^{76} - 128 q^{81} + 48 q^{86} - 40 q^{91} - 136 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.41419 0.00737473i 0.633877 0.633877i 1.99989 + 0.0208586i 0 −0.901100 + 0.891750i −0.524995 + 0.524995i −2.82808 0.0442468i 2.19640i 0
307.2 −1.40990 + 0.110380i 2.28932 2.28932i 1.97563 0.311250i 0 −2.97502 + 3.48041i 2.57361 2.57361i −2.75109 + 0.656902i 7.48197i 0
307.3 −1.37070 + 0.348126i 1.35566 1.35566i 1.75762 0.954349i 0 −1.38626 + 2.33014i −1.02862 + 1.02862i −2.07693 + 1.91999i 0.675627i 0
307.4 −1.35196 + 0.414984i −1.40270 + 1.40270i 1.65558 1.12208i 0 1.31429 2.47848i −1.55425 + 1.55425i −1.77262 + 2.20404i 0.935108i 0
307.5 −1.29808 0.561234i −0.556035 + 0.556035i 1.37003 + 1.45706i 0 1.03384 0.409713i 0.573665 0.573665i −0.960665 2.66029i 2.38165i 0
307.6 −1.20051 0.747512i 1.56869 1.56869i 0.882453 + 1.79479i 0 −3.05584 + 0.710615i 2.40449 2.40449i 0.282233 2.81431i 1.92156i 0
307.7 −1.19936 + 0.749357i −2.28062 + 2.28062i 0.876928 1.79750i 0 1.02629 4.44429i 1.63345 1.63345i 0.295216 + 2.81298i 7.40249i 0
307.8 −1.13692 0.841074i 1.03954 1.03954i 0.585190 + 1.91247i 0 −2.05621 + 0.307547i −2.88935 + 2.88935i 0.943215 2.66652i 0.838715i 0
307.9 −0.841074 1.13692i 1.03954 1.03954i −0.585190 + 1.91247i 0 −2.05621 0.307547i 2.88935 2.88935i 2.66652 0.943215i 0.838715i 0
307.10 −0.749357 + 1.19936i 2.28062 2.28062i −0.876928 1.79750i 0 1.02629 + 4.44429i 1.63345 1.63345i 2.81298 + 0.295216i 7.40249i 0
307.11 −0.747512 1.20051i 1.56869 1.56869i −0.882453 + 1.79479i 0 −3.05584 0.710615i −2.40449 + 2.40449i 2.81431 0.282233i 1.92156i 0
307.12 −0.561234 1.29808i −0.556035 + 0.556035i −1.37003 + 1.45706i 0 1.03384 + 0.409713i −0.573665 + 0.573665i 2.66029 + 0.960665i 2.38165i 0
307.13 −0.414984 + 1.35196i 1.40270 1.40270i −1.65558 1.12208i 0 1.31429 + 2.47848i −1.55425 + 1.55425i 2.20404 1.77262i 0.935108i 0
307.14 −0.348126 + 1.37070i −1.35566 + 1.35566i −1.75762 0.954349i 0 −1.38626 2.33014i −1.02862 + 1.02862i 1.91999 2.07693i 0.675627i 0
307.15 −0.110380 + 1.40990i −2.28932 + 2.28932i −1.97563 0.311250i 0 −2.97502 3.48041i 2.57361 2.57361i 0.656902 2.75109i 7.48197i 0
307.16 −0.00737473 1.41419i 0.633877 0.633877i −1.99989 + 0.0208586i 0 −0.901100 0.891750i 0.524995 0.524995i 0.0442468 + 2.82808i 2.19640i 0
307.17 0.00737473 + 1.41419i −0.633877 + 0.633877i −1.99989 + 0.0208586i 0 −0.901100 0.891750i −0.524995 + 0.524995i −0.0442468 2.82808i 2.19640i 0
307.18 0.110380 1.40990i 2.28932 2.28932i −1.97563 0.311250i 0 −2.97502 3.48041i −2.57361 + 2.57361i −0.656902 + 2.75109i 7.48197i 0
307.19 0.348126 1.37070i 1.35566 1.35566i −1.75762 0.954349i 0 −1.38626 2.33014i 1.02862 1.02862i −1.91999 + 2.07693i 0.675627i 0
307.20 0.414984 1.35196i −1.40270 + 1.40270i −1.65558 1.12208i 0 1.31429 + 2.47848i 1.55425 1.55425i −2.20404 + 1.77262i 0.935108i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.k.f 64
5.b even 2 1 inner 1000.2.k.f 64
5.c odd 4 2 inner 1000.2.k.f 64
8.d odd 2 1 inner 1000.2.k.f 64
40.e odd 2 1 inner 1000.2.k.f 64
40.k even 4 2 inner 1000.2.k.f 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.k.f 64 1.a even 1 1 trivial
1000.2.k.f 64 5.b even 2 1 inner
1000.2.k.f 64 5.c odd 4 2 inner
1000.2.k.f 64 8.d odd 2 1 inner
1000.2.k.f 64 40.e odd 2 1 inner
1000.2.k.f 64 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{32} + 277 T_{3}^{28} + 25958 T_{3}^{24} + 977049 T_{3}^{20} + 16834070 T_{3}^{16} + 132564949 T_{3}^{12} + 404639833 T_{3}^{8} + 317994992 T_{3}^{4} + 69488896 \) Copy content Toggle raw display
\( T_{7}^{32} + 645 T_{7}^{28} + 144130 T_{7}^{24} + 13350515 T_{7}^{20} + 478100250 T_{7}^{16} + 6538914875 T_{7}^{12} + 24095051650 T_{7}^{8} + 15165478125 T_{7}^{4} + \cdots + 2562890625 \) Copy content Toggle raw display