|k|         omega           v_g           v_p      |dv_g/c|    correction
------------------------------------------------------------------------------
  0.157080    0.15691819    0.99691733    0.99897223  3.082666e-03  2.054477e-03
  0.314159    0.31286893    0.98768834    0.99589274  1.231166e-02  8.197660e-03
  0.471239    0.46689073    0.97236992    0.99077290  2.763008e-02  1.836907e-02
  0.628319    0.61803399    0.95105652    0.98363164  4.894348e-02  3.246879e-02
  0.785398    0.76536686    0.92387953    0.97449536  7.612047e-02  5.035880e-02
  0.942478    0.90798100    0.89100652    0.96339776  1.089935e-01  7.186475e-02
  1.099557    1.04499713    0.85264016    0.95037976  1.473598e-01  9.677830e-02
  1.256637    1.17557050    0.80901699    0.93548928  1.909830e-01  1.248598e-01
  1.413717    1.29889610    0.76040597    0.91878104  2.395940e-01  1.558414e-01
  1.570796    1.41421356    0.70710678    0.90031632  2.928932e-01  1.894305e-01
  1.727876    1.52081193    0.64944805    0.88016268  3.505520e-01  2.253137e-01
  1.884956    1.61803399    0.58778525    0.85839369  4.122147e-01  2.631603e-01
  2.042035    1.70528033    0.52249856    0.83508860  4.775014e-01  3.026270e-01
  2.199115    1.78201305    0.45399050    0.81033196  5.460095e-01  3.433621e-01
  2.356194    1.84775907    0.38268343    0.78421330  6.173166e-01  3.850095e-01
  2.513274    1.90211303    0.30901699    0.75682673  6.909830e-01  4.272133e-01
  2.670354    1.94473984    0.23344536    0.72827049  7.665546e-01  4.696221e-01
  2.827433    1.97537668    0.15643447    0.69864659  8.435655e-01  5.118929e-01
  2.984513    1.99383467    0.07845910    0.66806030  9.215409e-01  5.536954e-01
  3.141593    2.00000000    0.00000000    0.63661977  1.000000e+00  5.947153e-01