Properties

Label 8013.2.a.b
Level $8013$
Weight $2$
Character orbit 8013.a
Self dual yes
Analytic conductor $63.984$
Analytic rank $0$
Dimension $106$
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] = [8013,2,Mod(1,8013)] 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("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [106] 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: \(63.9841271397\)
Analytic rank: \(0\)
Dimension: \(106\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9} - 3 q^{10} + 55 q^{11} - 109 q^{12} - 8 q^{13} + 27 q^{14} - 16 q^{15} + 111 q^{16} + 28 q^{17} + 15 q^{18}+ \cdots + 55 q^{99}+O(q^{100}) \) 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 −2.68955 −1.00000 5.23367 −0.143035 2.68955 0.169841 −8.69711 1.00000 0.384699
1.2 −2.68470 −1.00000 5.20764 3.29964 2.68470 2.53834 −8.61155 1.00000 −8.85857
1.3 −2.57806 −1.00000 4.64638 −0.00185853 2.57806 4.00944 −6.82252 1.00000 0.00479139
1.4 −2.57103 −1.00000 4.61019 2.04961 2.57103 −2.14469 −6.71087 1.00000 −5.26961
1.5 −2.57051 −1.00000 4.60751 −0.585495 2.57051 −2.49504 −6.70262 1.00000 1.50502
1.6 −2.45069 −1.00000 4.00587 0.123141 2.45069 −0.667497 −4.91577 1.00000 −0.301780
1.7 −2.44992 −1.00000 4.00213 3.65851 2.44992 −1.10640 −4.90506 1.00000 −8.96306
1.8 −2.43314 −1.00000 3.92015 4.32607 2.43314 4.27561 −4.67199 1.00000 −10.5259
1.9 −2.35891 −1.00000 3.56446 −3.22536 2.35891 −2.62511 −3.69041 1.00000 7.60832
1.10 −2.32341 −1.00000 3.39824 0.581135 2.32341 1.71987 −3.24868 1.00000 −1.35022
1.11 −2.30949 −1.00000 3.33375 0.518255 2.30949 1.93365 −3.08028 1.00000 −1.19691
1.12 −2.30696 −1.00000 3.32206 −3.03521 2.30696 1.29495 −3.04993 1.00000 7.00211
1.13 −2.24374 −1.00000 3.03438 2.67702 2.24374 −0.787815 −2.32089 1.00000 −6.00654
1.14 −2.16031 −1.00000 2.66695 −0.855629 2.16031 2.67269 −1.44083 1.00000 1.84843
1.15 −2.12375 −1.00000 2.51030 −3.37551 2.12375 −2.22714 −1.08375 1.00000 7.16873
1.16 −2.09214 −1.00000 2.37707 −2.25066 2.09214 −3.42799 −0.788875 1.00000 4.70871
1.17 −1.85225 −1.00000 1.43083 2.49048 1.85225 −3.91286 1.05424 1.00000 −4.61299
1.18 −1.84122 −1.00000 1.39011 −1.17426 1.84122 1.69712 1.12295 1.00000 2.16208
1.19 −1.83416 −1.00000 1.36415 −3.29261 1.83416 −0.883876 1.16624 1.00000 6.03918
1.20 −1.79051 −1.00000 1.20593 −0.580006 1.79051 0.129222 1.42179 1.00000 1.03851
See next 80 embeddings (of 106 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.106
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(2671\) \( -1 \)

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 8013.2.a.b ✓ 106
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8013.2.a.b ✓ 106 1.a even 1 1 trivial