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Properties

Label 8001.2.a.y
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [28,0,0,30,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55}+ \cdots - 8 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.72816 0 5.44285 0.297532 0 −1.00000 −9.39264 0 −0.811714
1.2 −2.71687 0 5.38141 −1.23671 0 −1.00000 −9.18686 0 3.35997
1.3 −2.47474 0 4.12433 −3.91787 0 −1.00000 −5.25716 0 9.69570
1.4 −2.27499 0 3.17556 3.81133 0 −1.00000 −2.67439 0 −8.67073
1.5 −2.02489 0 2.10018 2.65037 0 −1.00000 −0.202857 0 −5.36671
1.6 −1.99802 0 1.99208 −1.55092 0 −1.00000 0.0158280 0 3.09877
1.7 −1.55816 0 0.427866 −3.44623 0 −1.00000 2.44964 0 5.36979
1.8 −1.42730 0 0.0371861 2.54057 0 −1.00000 2.80152 0 −3.62615
1.9 −1.34710 0 −0.185322 0.124848 0 −1.00000 2.94385 0 −0.168183
1.10 −1.24740 0 −0.443996 1.34151 0 −1.00000 3.04864 0 −1.67339
1.11 −0.662421 0 −1.56120 −3.26980 0 −1.00000 2.35901 0 2.16599
1.12 −0.512239 0 −1.73761 1.76990 0 −1.00000 1.91455 0 −0.906610
1.13 −0.487315 0 −1.76252 0.708639 0 −1.00000 1.83353 0 −0.345330
1.14 −0.0958522 0 −1.99081 1.26637 0 −1.00000 0.382528 0 −0.121384
1.15 0.0958522 0 −1.99081 −1.26637 0 −1.00000 −0.382528 0 −0.121384
1.16 0.487315 0 −1.76252 −0.708639 0 −1.00000 −1.83353 0 −0.345330
1.17 0.512239 0 −1.73761 −1.76990 0 −1.00000 −1.91455 0 −0.906610
1.18 0.662421 0 −1.56120 3.26980 0 −1.00000 −2.35901 0 2.16599
1.19 1.24740 0 −0.443996 −1.34151 0 −1.00000 −3.04864 0 −1.67339
1.20 1.34710 0 −0.185322 −0.124848 0 −1.00000 −2.94385 0 −0.168183
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.y ✓ 28
3.b odd 2 1 inner 8001.2.a.y ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.y ✓ 28 1.a even 1 1 trivial
8001.2.a.y ✓ 28 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{28} - 43 T_{2}^{26} + 813 T_{2}^{24} - 8906 T_{2}^{22} + 62714 T_{2}^{20} - 297802 T_{2}^{18} + \cdots + 100 \) Copy content Toggle raw display
\( T_{5}^{28} - 77 T_{5}^{26} + 2556 T_{5}^{24} - 48052 T_{5}^{22} + 565977 T_{5}^{20} - 4371605 T_{5}^{18} + \cdots + 29584 \) Copy content Toggle raw display