K = 3 and two links with all couplings

Here are the eigenvalues for the constants:
{mub, muf, g2, kappas, kappaa, beta, lambda1, lambda2, lambda3, lambda4, lambda5} =
{0.1803, 0.362, 1, -0.323, 0.162, 0.688, -0.038, -0.09, 374, 0.158, 143}

K = 3 and two links with all couplings

In[143]:=
expr = N[generalMatrix[makeham, 3, 2, 0, 
        1, {0.1803, 0.362, 1, -0.323, 0.162, 0.688, -0.038, -0.09, 374, 0.158,
           143}, none]];

In[145]:=
Sort[Chop[Eigenvalues[expr]]]

Out[145]=
{0.400438, 0.411482, 0.411482, 0.42319, 1.46882, 1.47553, 1.47553, 1.47875, \
1.48206, 1.48429, 1.48429, 1.48567, 1.48797, 1.50733, 1.50733, 1.53345, \
1.64714, 1.66904, 1.67203, 1.67203, 1.67547, 1.67691, 1.67691, 1.70437, \
1.79909, 1.8029, 1.80603, 1.80603, 1.81608, 1.82488, 1.82488, 1.82732, \
1.83398, 1.83572, 1.83572, 1.85158, 2.18084, 2.19021, 2.19021, 2.20088, \
2.27654, 2.28849, 2.28849, 2.30289, 2.46224, 2.46224, 2.47218, 2.48466, \
2.48816, 2.49088, 2.49866, 2.49866, 2.50462, 2.50495, 2.50495, 2.50508, \
2.50624, 2.50624, 2.50997, 2.51879, 2.52396, 2.52865, 2.53783, 2.53783, \
2.60311, 2.60808, 2.60966, 2.60966, 2.66114, 2.66226, 2.66228, 2.66277, \
2.66277, 2.66524, 2.67069, 2.67587, 2.6776, 2.6776, 2.67985, 2.67985, 2.6885, \
2.6885, 2.69167, 2.69279, 2.69571, 2.69571, 2.69599, 2.72245, 2.79886, \
2.80081, 2.80081, 2.80924, 2.91087, 2.91125, 2.91125, 2.91313, 3.06809, \
3.06835, 3.08487, 3.08487, 3.08877, 3.08877, 3.10443, 3.10689, 3.12414, \
3.12512, 3.12512, 3.12567, 3.40825, 3.41019, 3.41138, 3.41138, 3.41236, \
3.41328, 3.41792, 3.41792, 3.42172, 3.42172, 3.42285, 3.43372, 3.70749, \
3.70851, 3.70851, 3.70968}

These are the diagonal matrix elements 

In[146]:=
diag = Table[expr[[i, i]], {i, Length[expr]}]

Out[146]=
{1.61516, 1.61516, 1.61516, 1.61516, 1.64536, 1.64536, 1.64536, 1.64536, \
1.61516, 1.61516, 1.61516, 1.61516, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.87204, 2.87204, 2.87204, 2.87204, \
2.87204, 2.87204, 2.87204, 2.87204, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.87204, 2.87204, 2.87204, 2.87204, \
2.87204, 2.87204, 2.87204, 2.87204, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.59188, 2.59188, 2.65109, 2.65109, \
2.65109, 2.65109, 2.59523, 2.59523, 2.65109, 2.65109, 2.59188, 2.59188, \
2.59523, 2.59523, 2.65109, 2.65109, 2.65109, 2.65109, 2.59523, 2.59523, \
2.59188, 2.59188, 2.65109, 2.65109, 2.59523, 2.59523, 2.65109, 2.65109, \
2.65109, 2.65109, 2.59188, 2.59188, 2.59188, 2.59188, 2.65109, 2.65109, \
2.65109, 2.65109, 2.59523, 2.59523, 2.65109, 2.65109, 2.59188, 2.59188, \
2.59523, 2.59523, 2.65109, 2.65109, 2.65109, 2.65109, 2.59523, 2.59523, \
2.59188, 2.59188, 2.65109, 2.65109, 2.59523, 2.59523, 2.65109, 2.65109, \
2.65109, 2.65109, 2.59188, 2.59188}

This is a list of the different elements of the matrix

In[144]:=
Union[Flatten[expr]]

Out[144]=
{-0.4976111526621536, -0.477464829275686, -0.164247901270836, -0.1193662073189215, 
  -0.06617266787514924, -0.05614937062046789, -0.021485917317405873, -0.016655883104453036, 
  -0.01323453357502985, -0.00999352986267182, -0.008327941552226518 - 0.008327941552226518*I, 
  -0.008327941552226518 + 0.008327941552226518*I, -0.007066492733885511, -0.00499676493133591, 
  -0.00499676493133591 - 0.00499676493133591*I, -0.00499676493133591 + 0.00499676493133591*I, 
  -0.0035332463669427556 - 0.0035332463669427556*I, -0.0035332463669427556 + 
   0.0035332463669427556*I, -0.002498382465667955 - 0.002498382465667955*I, 
  -0.002498382465667955 + 0.002498382465667955*I, 0., 0. - 0.00499676493133591*I, 
  0. + 0.00499676493133591*I, 0. - 0.007066492733885511*I, 0. + 0.007066492733885511*I, 
  0. - 0.00999352986267182*I, 0. + 0.00999352986267182*I, 0. - 0.01323453357502985*I, 
  0. + 0.01323453357502985*I, 0. - 0.016655883104453036*I, 0. + 0.016655883104453036*I, 
  0. - 0.05614937062046789*I, 0. + 0.05614937062046789*I, 0. - 0.06617266787514924*I, 
  0. + 0.06617266787514924*I, 0.002498382465667955 - 0.002498382465667955*I, 
  0.002498382465667955 + 0.002498382465667955*I, 0.0035332463669427556 - 
   0.0035332463669427556*I, 0.0035332463669427556 + 0.0035332463669427556*I, 
  0.00499676493133591, 0.00499676493133591 - 0.00499676493133591*I, 
  0.00499676493133591 + 0.00499676493133591*I, 0.005272589966034574, 
  0.007044682571126605 - 0.0017720926050920319*I, 0.007044682571126605 + 
   0.0017720926050920319*I, 0.007066492733885511, 0.007456568238798389, 
  0.008327941552226518 - 0.008327941552226518*I, 0.008327941552226518 + 
   0.008327941552226518*I, 0.00996268563470061 + 0.*I, 
  0.00996268563470061 - 0.0025061173959022207*I, 0.00996268563470061 + 
   0.0025061173959022207*I, 0.00999352986267182, 0.01242761373133065, 0.01323453357502985, 
  0.01660447605783435 - 0.004176862326503702*I, 0.01660447605783435 + 0.004176862326503702*I, 
  0.016655883104453036, 0.01988418197012904, 0.02078133838433805, 
  0.02656716169253496 - 0.0016707449306014807*I, 0.02656716169253496 + 
   0.0016707449306014807*I, 0.02656716169253496 - 0.006682979722405923*I, 
  0.02656716169253496 + 0.006682979722405923*I, 0.0377197215127792, 0.05614937062046789, 
  0.06617266787514924, 0.10554948999344792, 0.16792821028158453, 0.26387372498361983, 
  1.6151606547782247, 1.6453563656260766, 2.168277470950887, 2.5918843860495544, 
  2.5952266398544843, 2.6510900248797395, 2.8720431068561347}