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Properties

Label 9971.2.a.p
Level $9971$
Weight $2$
Character orbit 9971.a
Self dual yes
Analytic conductor $79.619$
Analytic rank $1$
Dimension $69$
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] = [9971,2,Mod(1,9971)] 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("9971.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9971, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9971 = 13^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9971.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [69,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.6188358554\)
Analytic rank: \(1\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - q^{2} - 6 q^{3} + 45 q^{4} + 5 q^{5} - 4 q^{6} + 5 q^{7} - 3 q^{8} + 35 q^{9} - 28 q^{10} - 3 q^{11} - 18 q^{12} - 15 q^{14} + 9 q^{15} + q^{16} - 17 q^{17} + 21 q^{18} + 6 q^{19} + 7 q^{20} - 5 q^{21}+ \cdots - 82 q^{99}+O(q^{100}) \) 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 −2.63225 −0.646104 4.92876 −1.44999 1.70071 2.11509 −7.70923 −2.58255 3.81674
1.2 −2.60410 −2.03172 4.78131 −1.66976 5.29080 0.861004 −7.24280 1.12789 4.34822
1.3 −2.58344 −1.13675 4.67415 2.63763 2.93672 1.41306 −6.90851 −1.70781 −6.81415
1.4 −2.55433 1.23249 4.52459 0.952133 −3.14819 −0.527189 −6.44863 −1.48096 −2.43206
1.5 −2.50382 −0.865121 4.26914 3.08872 2.16611 −3.86584 −5.68153 −2.25157 −7.73360
1.6 −2.34688 1.86620 3.50786 0.126814 −4.37976 3.65564 −3.53878 0.482711 −0.297617
1.7 −2.29570 −0.920733 3.27025 0.666308 2.11373 −3.60636 −2.91611 −2.15225 −1.52964
1.8 −2.14289 1.90706 2.59200 0.717460 −4.08662 −0.977198 −1.26859 0.636865 −1.53744
1.9 −2.12505 2.55679 2.51586 3.94324 −5.43332 0.119191 −1.09623 3.53717 −8.37960
1.10 −2.12157 −0.0963102 2.50104 −1.65241 0.204328 0.704961 −1.06299 −2.99072 3.50570
1.11 −1.87877 −2.94985 1.52978 −1.76790 5.54209 −0.844159 0.883435 5.70160 3.32147
1.12 −1.86507 2.16364 1.47850 −0.345844 −4.03536 2.98952 0.972629 1.68135 0.645024
1.13 −1.79830 −2.15842 1.23388 3.62466 3.88149 −2.02757 1.37772 1.65879 −6.51823
1.14 −1.76035 0.956916 1.09882 −1.88017 −1.68450 −2.26452 1.58638 −2.08431 3.30975
1.15 −1.65300 −0.637646 0.732419 1.75361 1.05403 −2.87232 2.09532 −2.59341 −2.89872
1.16 −1.58111 1.52634 0.499919 −4.39192 −2.41332 1.98971 2.37180 −0.670279 6.94413
1.17 −1.49082 −0.421763 0.222548 2.15567 0.628773 2.36089 2.64986 −2.82212 −3.21373
1.18 −1.40148 −1.54644 −0.0358423 −3.20979 2.16731 2.96757 2.85320 −0.608519 4.49847
1.19 −1.39033 2.82609 −0.0669779 0.979179 −3.92920 3.60286 2.87378 4.98679 −1.36138
1.20 −1.35135 1.72805 −0.173855 2.16552 −2.33520 −1.94864 2.93764 −0.0138532 −2.92637
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)
\(59\) \( +1 \)

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 9971.2.a.p ✓ 69
13.b even 2 1 9971.2.a.q yes 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9971.2.a.p ✓ 69 1.a even 1 1 trivial
9971.2.a.q yes 69 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{69} + T_{2}^{68} - 91 T_{2}^{67} - 89 T_{2}^{66} + 3937 T_{2}^{65} + 3769 T_{2}^{64} - 107779 T_{2}^{63} + \cdots - 889 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9971))\). Copy content Toggle raw display