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Properties

Label 5001.2.a.c
Level $5001$
Weight $2$
Character orbit 5001.a
Self dual yes
Analytic conductor $39.933$
Analytic rank $1$
Dimension $60$
CM no
Inner twists $1$

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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] = [5001,2,Mod(1,5001)] 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("5001.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5001, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5001 = 3 \cdot 1667 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5001.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9331860508\)
Analytic rank: \(1\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 13 q^{2} + 60 q^{3} + 43 q^{4} - 35 q^{5} - 13 q^{6} - 14 q^{7} - 36 q^{8} + 60 q^{9} - 6 q^{10} - 33 q^{11} + 43 q^{12} - 14 q^{13} - 42 q^{14} - 35 q^{15} + 17 q^{16} - 49 q^{17} - 13 q^{18} - 23 q^{19}+ \cdots - 33 q^{99}+O(q^{100}) \) 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 −2.76047 1.00000 5.62020 1.18179 −2.76047 1.32411 −9.99345 1.00000 −3.26230
1.2 −2.70522 1.00000 5.31821 −3.76387 −2.70522 3.88924 −8.97647 1.00000 10.1821
1.3 −2.61959 1.00000 4.86227 −2.98891 −2.61959 −0.679017 −7.49798 1.00000 7.82972
1.4 −2.58325 1.00000 4.67317 −2.67718 −2.58325 1.65487 −6.90546 1.00000 6.91582
1.5 −2.48932 1.00000 4.19671 2.35562 −2.48932 0.444118 −5.46833 1.00000 −5.86390
1.6 −2.47737 1.00000 4.13738 1.10906 −2.47737 −3.88940 −5.29509 1.00000 −2.74755
1.7 −2.41910 1.00000 3.85203 −1.96598 −2.41910 3.26154 −4.48024 1.00000 4.75589
1.8 −2.40692 1.00000 3.79326 0.777558 −2.40692 5.01268 −4.31624 1.00000 −1.87152
1.9 −2.19990 1.00000 2.83958 −3.02834 −2.19990 −3.42382 −1.84700 1.00000 6.66205
1.10 −2.08442 1.00000 2.34480 −1.40969 −2.08442 −2.24491 −0.718707 1.00000 2.93838
1.11 −2.06304 1.00000 2.25612 2.99778 −2.06304 −2.77406 −0.528389 1.00000 −6.18452
1.12 −1.98189 1.00000 1.92789 −0.0245117 −1.98189 2.32565 0.142912 1.00000 0.0485796
1.13 −1.95317 1.00000 1.81485 −4.08909 −1.95317 −1.54860 0.361619 1.00000 7.98667
1.14 −1.90625 1.00000 1.63378 2.66277 −1.90625 −0.681992 0.698098 1.00000 −5.07589
1.15 −1.68593 1.00000 0.842372 −0.865004 −1.68593 1.96855 1.95168 1.00000 1.45834
1.16 −1.63105 1.00000 0.660340 −3.63285 −1.63105 3.00495 2.18506 1.00000 5.92538
1.17 −1.62763 1.00000 0.649179 1.78502 −1.62763 −1.36296 2.19864 1.00000 −2.90535
1.18 −1.54512 1.00000 0.387410 3.11917 −1.54512 2.87358 2.49165 1.00000 −4.81951
1.19 −1.47715 1.00000 0.181982 1.00260 −1.47715 −2.39320 2.68549 1.00000 −1.48099
1.20 −1.40185 1.00000 −0.0348269 −1.96409 −1.40185 −4.72155 2.85251 1.00000 2.75335
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(1667\) \( -1 \)

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 5001.2.a.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5001.2.a.c ✓ 60 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 13 T_{2}^{59} + 3 T_{2}^{58} - 664 T_{2}^{57} - 2183 T_{2}^{56} + 14182 T_{2}^{55} + \cdots + 8551 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5001))\). Copy content Toggle raw display