Determine a positive semidefinite 5*5 matrix

I’d like to find the real parameters $left{a_{14},c_6,c_8,c_{10},c_{12},c_{13},c_{14},c_{15},alpha right}$ in M, which is a $5times 5$ real symmetric matrix, such that M is positive semidefinite. I assume that $alphage 1$. My idea is to use Descartes’ rule of sign (https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs) to determine the sign of the real roots of the corresponding characteristic polynomial $p(lambda)$.

However, the coefficients of $p(lambda)$ are rather complicated. Any reference, suggestion, idea, or comment is welcome. Thank you!

My code for $M$:

MatrixForm[{{(-7 + α)*Subscript[c, 15], (1/2)*((-6 + α)*Subscript[c, 14] + 7*Subscript[c, 15]), 
    (1/2)*((-5 + α)*Subscript[c, 13] + Subscript[c, 14]), Subscript[a, 14], 
    (1/2)*((-4 + α)*(-3 + α)*(-2 + α)*(-1 + α) + Subscript[c, 13])}, 
   {(1/2)*((-6 + α)*Subscript[c, 14] + 7*Subscript[c, 15]), -2*Subscript[a, 14] + (-5 + α)*Subscript[c, 12] + 
     5*Subscript[c, 14], (1/2)*(-4 + α)*Subscript[c, 10] + Subscript[c, 12] + 2*Subscript[c, 13], 
    (1/2)*((-4 + α)*Subscript[c, 8] + 3*Subscript[c, 12]), 5*(-3 + α)*(-2 + α)*(-1 + α) + Subscript[c, 10]/2}, 
   {(1/2)*((-5 + α)*Subscript[c, 13] + Subscript[c, 14]), (1/2)*(-4 + α)*Subscript[c, 10] + Subscript[c, 12] + 
     2*Subscript[c, 13], (-3 + α)*Subscript[c, 6] + Subscript[c, 10], 
    (1/4)*((-(-3 + α))*Subscript[c, 6] + 6*Subscript[c, 8] + 4*Subscript[c, 10]), 6*(-2 + α)*(-1 + α) + Subscript[c, 6]}, 
   {Subscript[a, 14], (1/2)*((-4 + α)*Subscript[c, 8] + 3*Subscript[c, 12]), 
    (1/4)*((-(-3 + α))*Subscript[c, 6] + 6*Subscript[c, 8] + 4*Subscript[c, 10]), 
    (-3 + α)*(-2 + α)*(-1 + α) + Subscript[c, 8], (1/4)*(26*(-2 + α)*(-1 + α) - Subscript[c, 6])}, 
   {(1/2)*((-4 + α)*(-3 + α)*(-2 + α)*(-1 + α) + Subscript[c, 13]), 5*(-3 + α)*(-2 + α)*(-1 + α) + Subscript[c, 10]/2, 
    6*(-2 + α)*(-1 + α) + Subscript[c, 6], (1/4)*(26*(-2 + α)*(-1 + α) - Subscript[c, 6]), 8*(-1 + α)}}]

PositiveSemidefiniteMatrixQ[  m /. {
  α -> 2,
  Subscript[a, 14] -> 0,
  Subscript[c, 6] -> 0,
  Subscript[c, 8] -> 0,
  Subscript[c, 10] -> 0,
  Subscript[c, 12] -> 0,
  Subscript[c, 13] -> 0,
  Subscript[c, 14] -> 0,
  Subscript[c, 15] -> 0}
]
(* True *)

evals = Eigenvalues[m];
{err, result} = NMaximize[Min[evals], Variables[m]]
(* {0.00105605, {α -> 1.0778, Subscript[a, 14] -> 0.174339, 
  Subscript[c, 6] -> -0.258389, Subscript[c, 8] -> 0.125641, 
  Subscript[c, 10] -> 0.373052, Subscript[c, 12] -> -0.264517, 
  Subscript[c, 13] -> -0.0915956, Subscript[c, 14] -> 0.133419, 
  Subscript[c, 15] -> -0.0302445}} *)

PositiveSemidefiniteMatrixQ[m /. result]
(* True *)