Fourier transform of a time series

Question

I have this question:

Create a time series of the function $\sin(7*\sin(140*t))$ with a sampling rate of $1000Hz$ for a time duration of $2.5s$. Fourier transform this time series and identify the absolute value (amplitude) of the largest Fourier component in the range $0$ Hz to $450$ Hz.

My code so far is:

ClearAll["Global`*"]
numpt = 2500; dt = 400; T = dt numpt;
data = Table[Sin[7*Sin[140*t]], {t, 0, T - dt, dt}];
fdata = Abs[Fourier[data]];
MaxValue[fdata]]

However i am struggling to find the max value in the range.

Answer 1

ClearAll["Global`*"]
sampleRate = 1000.; duration = 2.5; dt = 1/sampleRate;
data = Table[Sin[7*Sin[140*t]], {t, 0, duration, dt}];
fdata = Abs[Fourier[data]];
todecibels[pwr_] := 10 Log10[pwr]
peaks = {#[[1]]/duration, todecibels[#[[2]]^2]} & /@ FindPeaks[fdata];
Periodogram[data, SampleRate -> sampleRate, 
 PlotRange -> {{0, 600}, Full}, ScalingFunctions -> "dB", 
 Epilog -> {Red, Point[peaks]}]

peaks

Your peaks are at these frequencies (in Hz), and amplitudes in dB:

{{0.4, -29.9564}, {22.8, -12.4852}, {67.2, 17.9721}, {112., 22.7784},
 {156.4, 21.26}, {201.2, 6.14708}, {245.6, -7.45039}, {290.4, -31.4171},
 {334.8, -43.1615}, {666.4, -43.1615}, {710.8, -31.4171}, {755.6, -7.45039}, 
 {800., 6.14708}, {844.8, 21.26}, {889.2, 22.7784}, {934., 17.9721},
 {978.4, -12.4852}, {1000.4, -29.9564}}