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Properties

Label 8001.2.a.z
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 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 −2.69601 0 5.26849 3.35934 0 −1.00000 −8.81189 0 −9.05682
1.2 −2.49464 0 4.22324 −0.373762 0 −1.00000 −5.54618 0 0.932401
1.3 −2.44637 0 3.98471 1.74324 0 −1.00000 −4.85534 0 −4.26460
1.4 −2.41110 0 3.81340 −2.91376 0 −1.00000 −4.37229 0 7.02536
1.5 −2.23843 0 3.01059 −0.715002 0 −1.00000 −2.26213 0 1.60049
1.6 −1.96743 0 1.87078 2.35101 0 −1.00000 0.254227 0 −4.62544
1.7 −1.66842 0 0.783626 −3.76494 0 −1.00000 2.02942 0 6.28150
1.8 −1.63487 0 0.672794 0.107826 0 −1.00000 2.16981 0 −0.176281
1.9 −1.56027 0 0.434445 1.69832 0 −1.00000 2.44269 0 −2.64985
1.10 −1.18013 0 −0.607291 4.43826 0 −1.00000 3.07694 0 −5.23773
1.11 −0.959867 0 −1.07866 −2.40466 0 −1.00000 2.95510 0 2.30815
1.12 −0.942656 0 −1.11140 −2.02754 0 −1.00000 2.93298 0 1.91127
1.13 −0.904420 0 −1.18202 −0.974726 0 −1.00000 2.87789 0 0.881562
1.14 −0.903971 0 −1.18284 2.97865 0 −1.00000 2.87719 0 −2.69261
1.15 −0.242500 0 −1.94119 2.79671 0 −1.00000 0.955741 0 −0.678204
1.16 −0.203292 0 −1.95867 −2.16833 0 −1.00000 0.804765 0 0.440803
1.17 0.203292 0 −1.95867 2.16833 0 −1.00000 −0.804765 0 0.440803
1.18 0.242500 0 −1.94119 −2.79671 0 −1.00000 −0.955741 0 −0.678204
1.19 0.903971 0 −1.18284 −2.97865 0 −1.00000 −2.87719 0 −2.69261
1.20 0.904420 0 −1.18202 0.974726 0 −1.00000 −2.87789 0 0.881562
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
\(3\) \(1\)
\(7\) \(1\)
\(127\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.z ✓ 32
3.b odd 2 1 inner 8001.2.a.z ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.z ✓ 32 1.a even 1 1 trivial
8001.2.a.z ✓ 32 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{32} - 47 T_{2}^{30} + 991 T_{2}^{28} - 12396 T_{2}^{26} + 102523 T_{2}^{24} - 591710 T_{2}^{22} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{5}^{32} - 98 T_{5}^{30} + 4277 T_{5}^{28} - 110184 T_{5}^{26} + 1870977 T_{5}^{24} - 22127702 T_{5}^{22} + \cdots + 7929856 \) Copy content Toggle raw display