↑

Properties

Label 8002.2.a.e
Level $8002$
Weight $2$
Character orbit 8002.a
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $77$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8002,2,Mod(1,8002)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8002.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8002, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [77,-77,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(77\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19}+ \cdots + 63 q^{99}+O(q^{100}) \) 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 −1.00000 −3.18344 1.00000 −0.604370 3.18344 1.50072 −1.00000 7.13428 0.604370
1.2 −1.00000 −3.14716 1.00000 −2.17339 3.14716 4.77090 −1.00000 6.90459 2.17339
1.3 −1.00000 −3.14624 1.00000 3.08369 3.14624 2.61077 −1.00000 6.89883 −3.08369
1.4 −1.00000 −2.93461 1.00000 1.73622 2.93461 −0.969734 −1.00000 5.61192 −1.73622
1.5 −1.00000 −2.91842 1.00000 3.04954 2.91842 3.39338 −1.00000 5.51716 −3.04954
1.6 −1.00000 −2.69277 1.00000 0.930791 2.69277 −2.75733 −1.00000 4.25101 −0.930791
1.7 −1.00000 −2.62668 1.00000 0.854809 2.62668 1.79286 −1.00000 3.89945 −0.854809
1.8 −1.00000 −2.61328 1.00000 −3.49569 2.61328 −0.478818 −1.00000 3.82924 3.49569
1.9 −1.00000 −2.50919 1.00000 −3.69214 2.50919 −2.32427 −1.00000 3.29603 3.69214
1.10 −1.00000 −2.45617 1.00000 0.358646 2.45617 −1.91182 −1.00000 3.03279 −0.358646
1.11 −1.00000 −2.42114 1.00000 −1.19106 2.42114 −3.77401 −1.00000 2.86190 1.19106
1.12 −1.00000 −2.37610 1.00000 −3.32970 2.37610 3.18362 −1.00000 2.64585 3.32970
1.13 −1.00000 −2.19428 1.00000 0.0495220 2.19428 1.71879 −1.00000 1.81487 −0.0495220
1.14 −1.00000 −2.17233 1.00000 −1.31184 2.17233 −2.50290 −1.00000 1.71901 1.31184
1.15 −1.00000 −2.13252 1.00000 2.05525 2.13252 −1.57763 −1.00000 1.54765 −2.05525
1.16 −1.00000 −1.85869 1.00000 −0.435894 1.85869 1.03466 −1.00000 0.454718 0.435894
1.17 −1.00000 −1.73843 1.00000 4.07186 1.73843 −2.38793 −1.00000 0.0221381 −4.07186
1.18 −1.00000 −1.70235 1.00000 4.07630 1.70235 1.03752 −1.00000 −0.101993 −4.07630
1.19 −1.00000 −1.65847 1.00000 −0.802626 1.65847 −0.0977002 −1.00000 −0.249472 0.802626
1.20 −1.00000 −1.59889 1.00000 −0.0810696 1.59889 5.11538 −1.00000 −0.443564 0.0810696
See all 77 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.77
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(4001\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8002.2.a.e ✓ 77
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8002.2.a.e ✓ 77 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{77} - 10 T_{3}^{76} - 101 T_{3}^{75} + 1309 T_{3}^{74} + 4091 T_{3}^{73} - 81236 T_{3}^{72} + \cdots - 1136453792 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\). Copy content Toggle raw display