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Properties

Label 8001.2.a.x
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $22$
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: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 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 −2.37730 0 3.65154 2.39021 0 1.00000 −3.92621 0 −5.68223
1.2 −2.32723 0 3.41602 1.82757 0 1.00000 −3.29542 0 −4.25318
1.3 −2.21969 0 2.92701 0.739015 0 1.00000 −2.05767 0 −1.64038
1.4 −1.94436 0 1.78053 −1.34248 0 1.00000 0.426737 0 2.61025
1.5 −1.69586 0 0.875937 −2.42277 0 1.00000 1.90625 0 4.10867
1.6 −1.56609 0 0.452631 −2.71234 0 1.00000 2.42332 0 4.24776
1.7 −0.775659 0 −1.39835 −0.833166 0 1.00000 2.63596 0 0.646253
1.8 −0.691771 0 −1.52145 −0.236679 0 1.00000 2.43604 0 0.163728
1.9 −0.684852 0 −1.53098 −0.182709 0 1.00000 2.41820 0 0.125129
1.10 −0.446366 0 −1.80076 2.33773 0 1.00000 1.69653 0 −1.04348
1.11 −0.384540 0 −1.85213 3.33517 0 1.00000 1.48130 0 −1.28251
1.12 0.384540 0 −1.85213 −3.33517 0 1.00000 −1.48130 0 −1.28251
1.13 0.446366 0 −1.80076 −2.33773 0 1.00000 −1.69653 0 −1.04348
1.14 0.684852 0 −1.53098 0.182709 0 1.00000 −2.41820 0 0.125129
1.15 0.691771 0 −1.52145 0.236679 0 1.00000 −2.43604 0 0.163728
1.16 0.775659 0 −1.39835 0.833166 0 1.00000 −2.63596 0 0.646253
1.17 1.56609 0 0.452631 2.71234 0 1.00000 −2.42332 0 4.24776
1.18 1.69586 0 0.875937 2.42277 0 1.00000 −1.90625 0 4.10867
1.19 1.94436 0 1.78053 1.34248 0 1.00000 −0.426737 0 2.61025
1.20 2.21969 0 2.92701 −0.739015 0 1.00000 2.05767 0 −1.64038
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.x ✓ 22
3.b odd 2 1 inner 8001.2.a.x ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.x ✓ 22 1.a even 1 1 trivial
8001.2.a.x ✓ 22 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{22} - 27 T_{2}^{20} + 307 T_{2}^{18} - 1912 T_{2}^{16} + 7111 T_{2}^{14} - 16187 T_{2}^{12} + \cdots - 16 \) Copy content Toggle raw display
\( T_{5}^{22} - 42 T_{5}^{20} + 737 T_{5}^{18} - 7048 T_{5}^{16} + 40077 T_{5}^{14} - 138490 T_{5}^{12} + \cdots - 64 \) Copy content Toggle raw display