Just another hack: I applied my analytical description to the problem-definition.
I searched for a fixpoint t for the base-parameter b=I, such that t^(1/t)=b=I
I found:
t= 0.438282936727 + 0.360592471871*I
Check:
t^(1/t) = -4.98388377612 E-45 + 1.00000000000*I ~ 0.0 + 1.0*I
is good.
Then u= log(t):
u = log(t) = -0.566417330285 + 0.688453227108*I
Hypothesis: Eigenvalues are consecutive powers of u , beginning at u^0 .
We'll find such powers in the empirical set of eigenvalues with more or less good approximations
Code:
. 1.00000000000 (u^0)
-0.566417336767+0.688453222928*I (u^1)
-0.153139121702-0.779903738266*I (u^2)
0.623667274970+0.336312783860*I (u^3)
-0.584898170382+0.238979921688*I (u^4)
here approximations become worse:
0.166722386790-0.538414193535*I
0.271875740531+0.406308595179*I
-0.434207067913-0.0300570687829*I
0.253868227388-0.289262673366*I
0.0621911678425+0.0678908098416*I
The full set of eigenvalues, as reported before, but reordered are
\( \hspace{24}
\begin{matrix} {rrr}
1.00000000000 \\
-0.566417336767+0.688453222928*I \\
-0.153139121702-0.779903738266*I \\
0.623667274970+0.336312783860*I \\
-0.584898170382+0.238979921688*I \\
0.166722386790-0.538414193535*I \\
0.271875740531+0.406308595179*I \\
-0.434207067913-0.0300570687829*I \\
0.253868227388-0.289262673366*I \\
0.0621911678425+0.0678908098416*I \\
---- \\
-5.47974865102E17-2.37151639248E18*I \\
-1.40947398634E16+1.11407483052E16*I \\
2.46479084446E14+1.28514676152E14*I \\
558111745700.-6.74160002940E12*I \\
-220611525960.+65636453372.0*I \\
6058523291.41+8279445712.41*I \\
337747205.316-464618483.215*I \\
-36794031.6180-13535297.2580*I \\
-358363.072707+3168208.25176*I \\
302209.998163-27260.0213372*I \\
-8395.97431462-32222.7753783*I \\
-3862.15905348+1553.24166668*I \\
267.408153438+522.463483061*I \\
79.9487788351-46.6197593935*I \\
-8.44497709779-13.8323341991*I \\
-2.69468450558+1.59521959157*I \\
0.327850649567+0.623039459463*I \\
---\\
0.000214033167476+0.00585473360451*I \\
-0.000171842783002+0.000232593700346*I \\
-0.00000000174878768422-0.00000000220727624373*I \\
-0.000000233830386869-0.0000000768729721930*I \\
-0.0000100176583851+0.00000302704107609*I
\end{matrix} \)
The eigenvalues according to my hypothese (still not adapted to the problem of complex values) should be:
\( \hspace{24}
\begin{matrix} {rrr}
1 \\
-0.566417330285+0.688453227108*I \\
-0.153139253867-0.779903677850*I \\
0.623667931186+0.336321745566*I \\
-0.584798115648+0.238867734628*I \\
0.166790524665-0.537904974464*I \\
0.275849371850+0.419506174539*I \\
-0.445056244417-0.0477061771754*I \\
0.284931041419-0.279378802200*I \\
0.0309493581638+0.354406690248*I \\
-0.261522682435-0.179434905821*I \\
0.271663519562-0.0784110943693*I \\
-0.0998925545269+0.231441029468*I \\
-0.102755449572-0.199863561558*I \\
0.195799181354+0.0424638640984*I \\
-0.140138433849+0.110746309732*I \\
0.00313318324561-0.159207386122*I \\
0.107832149466+0.0923348727257*I \\
-0.124646239322+0.0219373191843*I \\
0.0554989719204-0.0982387834742*I \\
0.0361972280012+0.0938525957857*I \\
-0.0851158596893-0.0282396383155*I \\
0.0676527681408-0.0426028677382*I \\
-0.00898961853835+0.0707067691561*I
\end{matrix}
\)
Gottfried