d = 
      3

c[d_] = (4/d) (1/2) 
        (1/((((4  π))^(d/2)) Gamma[d/2]))

n = 
      2

V[ρ_] = (∑_(k = 1)^n) (Subscript[λ, 2  
                k]) (((ρ))^k)

Vdot[ρ_, 
        d_] = -d V[ρ] + (d - 2) ρ V'[ρ] + 
        c[d] (1/(1 + 2  V'[ρ] + 4  ρ V''[ρ]))

β[i_, d_] = 
        Simplify[(1/(i!)) Derivative[i, 0][Vdot][0, 
              d]];

eqns = 
        Table[Simplify[β[i, d]] == 0, {i, 1, n}];

var = Table[(Subscript[λ, 2  i]), {i, 1, n}]

fp = Solve[eqns, var];

N[fp]

wf = 
        Extract[fp, {2}];

N[wf]

M = 
        N[Table[D[β[j, d], (Subscript[λ, 2  i])], {i, 1, n}, {j, 1, n}] /. 
            wf];

MatrixForm[M]

e = N[Eigenvalues[M]]

-1/e

β[1, d]