relationship between modulation index and carrier-suppressed condition in fm
The answer to this fills whole chapters of some large textbooks[1], but to summarize a lengthy discussion involving Bessel functions of the first kind, the carrier component vanishes for M=2.405, 5.520, 8.653, 11.79, 14.93, ... Mathematically, these are the zeros of J0(M).

[1]: Cook &amp; Liff, Frequency Modulation Receivers, Prentice Hall, 1968, Fig 1-19, but you will find the same diagram in numerous texts.