Problem with symmetry and anti-symmetry of the fourier coefficients

let's say that I have a set of data

EchoHannatoS1 = {{- 0.010000000000005116`, 0.`}, {0.010000000000005116`, -0.0016729662767088105`}, {0.030000000000001137`, -0.00624162900299196`}, {0.04999999999999716`, -0.012853512800774878`}, {0.06999999999999318`, -0.021545951369554405`}, { 0.09000000000000341`, -0.031181593750000076`}, {0.10999999999999943`, -0.040817236130445744`}, {0.12999999999999545`, -0.05189135257766407`}, {0.15000000000000568`, -0.05912155849700829`}, {0.1700000000000017`, -0.0637636364488487`}, {0.18999999999999773`, -0.06536318750000025 `}, {0.20999999999999375`, -0.06136318750000025`}, {0.23000000000000398`, -0.055363187500000244`}, {0.25`, -0.04836318750000013`}, {0.269999999999996`, -0.04336318750000023`}, {0.29000000000000625`, -0.03936318750000023`}, {0.3100000000000023} `, -0.03936318750000023`}, {0.3299999999999983`, -0.03936318750000023`}, {0.3499999999999943`, -0.04336318750000023`}, {0.37000000000000455`, -0.04736318750000024`}, {0.39000000000000057`, -0.05336318750000024`}, {0.4099999999999966`, -0.057363187500000246` }, {0.4300000000000068`, -0. 05936318750000025`}, {0.45000000000000284`, -0.05936318750000025`}, {0.46999999999999886`, -0.05936318750000025`}, {0.4899999999999949`, -0.057363187500000246`}, {0.5100000000000051`, -0.05436318750000013`}, {0.5300000000000011`, -0.05036318750000013`}, { 0.5499999999999972`, -0.04736318750000024`}, {0.5699999999999932`, -0.045363187500000235`}, {0.5900000000000034`, -0.04336318750000023`}, {0.6099999999999994`, -0.04136318750000023`}, {0.6299999999999955`, -0.03836318750000012`}, {0.6500000000000057`, -0.034363187500000114 `}, {0.6700000000000017`, -0.03136318750000022`}, {0.6899999999999977`, -0.026363187500000107`}, {0.7099999999999937`, -0.023363187500000215`}, {0.730000000000004`, -0.0203631875000001`}, {0.75`, -0.0183631875000001`}, {0.769999999999996} `, -0.0183631875000001`}, {0.7900000000000063`, -0.0183631875000001`}, {0.8100000000000023`, -0.019363187500000212`}, {0.8299999999999983`, -0.023363187500000215`}, {0.8499999999999943`, -0.02836318750000011`}, {0.8700000000000045`, -0.034363187500000114` }, {0.8900000000000006`, - 0.04236318750000012`}, {0.9099999999999966`, -0.05036318750000013`}, {0.9300000000000068`, -0.057363187500000246`}, {0.9500000000000028`, -0.06436318750000014`}, {0.9699999999999989`, -0.06636318750000014`}, {0.9899999999999949`, -0.06636318750000014`}, { 1.0100000000000051`, -0.06636318750000014`}, {1.0300000000000011`, -0.06236318750000014`}, {1.0499999999999972`, -0.057363187500000246`}, {1.0699999999999932`, -0.05136318750000024`}, {1.0900000000000034`, -0.045363187500000235`}, {1.1099999999999994`, -0.04036318750000012 `}, {1.1299999999999955`, -0.036363187500000116`}, {1.1500000000000057`, -0.035363187500000226`}, {1.1700000000000017`, -0.035363187500000226`}, {1.1899999999999977`, -0.035363187500000226`}, {1.2099999999999937`, -0.03836318750000012`}, {1.230000000000004} `, -0.04436318750000012`}, {1.25`, -0.05136318750000024`}, {1.269999999999996`, -0.058363187500000135`}, {1.2900000000000063`, -0.06436318750000014`}, {1.3100000000000023`, -0.06736318750000025`}, {1.3299999999999983`, -0.06736318750000025` }, {1.349999999999994} 3`, -0.06736318750000025`}, {1.3700000000000045`, -0.06636318750000014`}, {1.3900000000000006`, -0.05936318750000025`}, {1.4099999999999966`, -0.05136318750000024`}, {1.4300000000000068`, -0.04336318750000023`}, {1.4500000000000028`, -0.03736318750000023 `}, {1.4699999999999989`, -0.03136318750000022`}, {1.4899999999999949`, -0.02936318750000022`}, {1.5100000000000051`, -0.02736318750000022`}, {1.5300000000000011`, -0.02736318750000022`}, {1.5499999999999972`, -0.02736318750000022`}, {1.5699999999999932} `, -0.026363187500000107`}, {1.5900000000000034`, -0.024363187500000105`}, {1.6099999999999994`, -0.022363187500000103`}, {1.6299999999999955`, -0.019363187500000212`}, {1.6500000000000057`, -0.019363187500000212`}, {1.6700000000000017`, -0.019363187500000212` }, {1.6899999999999977`, -0.022363187500000103`}, {1.7099999999999937`, -0.024363187500000105`}, {1.730000000000004`, -0.02736318750000022`}, {1.75`, -0.02736318750000022`}, {1.769999999999996`, -0.02736318750000022`}, {1.7900000000000063`} , -0.023363187500000215`}, {1.8100000000 000023`, -0.0183631875000001`}, {1.8299999999999983`, -0.010363187500000315`}, {1.8499999999999943`, -0.005363187500000199`}, {1.8700000000000045`, -0.0023631875000003078`}, {1.8900000000000006`, -0.0023631875000003078`}, {1.9099999999999966`, -0.0023631875000003078 `}, {1.9300000000000068`, -0.0033631875000001976`}, {1.9500000000000028`, -0.006363187500000311`}, {1.9699999999999989`, -0.010363187500000315`}, {1.9899999999999949`, -0.014363187500000318`}, {2.010000000000005`, -0.01736318750000021`}, {2.030000000000001} `, -0.01736318750000021`}, {2.049999999999997`, -0.01736318750000021`}, {2.069999999999993`, -0.014363187500000318`}, {2.0900000000000034`, -0.009363187500000203`}, {2.1099999999999994`, -0.0033631875000001976`}, {2.1299999999999955`, 0.0026368124999998077`} {2.1500000000000057`, 0.007636812499999701`}, {2.1700000000000017`, 0.007636812499999701`}, {2.1899999999999977`, 0.007636812499999701`}, {2.2099999999999937`, 0.007636812499999701`}, {2.230000000000004`, 0.0046368124999998095`}, {2.25`, 0.0006368124999998059`}, { 2.269999999999996`, -0.0033631875000001976`}, {2.2900000000000063`, -0.006363187500000311`}, {2.3100000000000023`, -0.009363187500000203`}, {2.3299999999999983`, -0.009363187500000203`}, {2.3499999999999943`, -0.009363187500000203`}, {2.3700000000000045`, -0.009363187500000203 `}, {2.3900000000000006`, -0.007363187500000191`}, {2.4099999999999966`, -0.006363187500000311`}, {2.430000000000007`, -0.005363187500000199`}, {2.450000000000003`, -0.005363187500000199`}, {2.469999999999999`, -0.005363187500000199`}, {2.489999999999995} `, -0.006363187500000311`}, {2.510000000000005`, -0.008363187500000313`}, {2.530000000000001`, -0.013363187500000206`}, {2.549999999999997`, -0.01736318750000021`}, {2.569999999999993`, -0.021363187500000214`}, {2.5900000000000034`, -0.021363187500000214` }, {2.6099999999999994`, -0.021363187500000214`}, {2.6299999999999955`, -0.019363187500000212`}, {2.6500000000000057`, -0.015363187500000208`}, {2.6700000000000017`, -0.009363187500000203`}, {2.6899999999999977`, -0.005363187500000199`}, {2.7099999999999937` , -0.0023631875000003078`}, {2.730000000000004`, -0.0013631875000001958`}, {2.75`, -0.0013631875000001958`}, {2.769999999999996`, -0.0013631875000001958`}, {2.7900000000000063`, -0.0033631875000001976`}, {2.8100000000000023`, -0.0033631875000001976`} , {2.8299999999999983`, -0.0033631875000001976`}, {2.8499999999999943`, -0.0033631875000001976`}, {2.8700000000000045`, -0.0033631875000001976`}, {2.8900000000000006`, -0.0033631875000001976`}, {2.9099999999999966`, -0.0033631875000001976`}, {2.930000000000007`, -0.0033631875000001976`}, {2.950000000000003`, -0.0023631875000003078`}, {2.969999999999999`, 0.0016368124999996958`}, {2.989999999999995`, 0.006636812499999811`}, {3.010000000000005`, 0.012636812499999817`}, {3.030000000000001`, 0.015636812499999708`}, {3.049999999999997`, 0.018636812499999822`}, {3.069999999999993`, 0.018636812499999822`}, {3.0900000000000034`, 0.018636812499999822`}, {3.1099999999999994`, 0.01663681249999982`}, {3.1299999999999955`, 0.01663681249999982`}, {3.1500000000000057`, 0.01663681249999982`}, {3.17000} 00000000017`, 0.01763681249999971`}, {3.1899999999999977`, 0.020636812499999824`}, {3.2099999999999937`, 0.025636812499999717`}, {3.230000000000004`, 0.02963681249999972`}, {3.25`, 0.035636812499999726`}, {3.269999999999996`, 0.04063681249999984`}, {3.2900000000000063` , 0.044636812499999845`}, {3.3100000000000023`, 0.04663681249999985`}, {3.3299999999999983`, 0.04663681249999985`}, {3.3499999999999943`, 0.04663681249999985`}, {3.3700000000000045`, 0.04263681249999984`}, {3.3900000000000006`, 0.03763681249999973`}, {3.4099999999999966`, .032636812499999834 '}, {3.430000000000007`, 0.02663681249999983`}, {3.450000000000003`, 0.019636812499999712`}, {3.469999999999999`, 0.014636812499999818`}, {3.489999999999995`, 0.007636812499999701`}, {3.510000000000005`, 0.0046368124999998095`}, {3.530000000000001`, 0.0016368124999996958`} , {3.549999999999997`, 0.0016368124999996958`}, {3.569999999999993`, 0.0016368124999996958`}, {3.5900000000000034`, 0.0026368124999998077`}, {3.6099999999999994`, 0.0036368124999996976`}, {3.6299999999999955`, 0. 005636812499999699`}, {3.6500000000000057`, 0.005636812499999699`}, {3.6700000000000017`, 0.005636812499999699`}, {3.6899999999999977`, 0.005636812499999699`}, {3.7099999999999937`, 0.006636812499999811`}, {3.730000000000004`, 0.009636812499999703`}, {3.75`, 0.013636812499999706` }, {3.769999999999996`, 0.018636812499999822`}, {3.7900000000000063`, 0.022636812499999825`}, {3.8100000000000023`, 0.02763681249999972`}, {3.8299999999999983`, 0.02963681249999972`}, {3.8499999999999943`, 0.03163681249999972`}, {3.8700000000000045`, 0.03163681249999972`}, {3.8900000000000006`, 0.03163681249999972`}, {3.9099999999999966`, 0.032636812499999834`}, {3.930000000000007`, 0.033636812499999724`}, {3.950000000000003`, 0.03663681249999984`}, {3.969999999999999`, 0.04163681249999973`}, {3.989999999999995`, 0.045636812499999735`}, {4.010000000000005 `, 0.05063681249999985`}, {4.030000000000001`, 0.05363681249999974`}, {4.049999999999997`, 0.057636812499999746`}, {4.069999999999993`, 0.05963681249999975`}, {4.090000000000003`, 0.06063681249999986`}, {4. 109999999999999`, 0.06163681249999975`}, {4.1299999999999955`, 0.06163681249999975`}, {4.150000000000006`, 0.06163681249999975`}, {4.170000000000002`, 0.06163681249999975`}, {4.189999999999998`, 0.05963681249999975`}, {4.209999999999994`, 0.056636812499999856`}, {4.230000000000004` , 0.054636812499999854`}, {4.25`, 0.05063681249999985`}, {4.269999999999996`, 0.04863681249999985`}, {4.290000000000006`, 0.04863681249999985`}, {4.310000000000002`, 0.04863681249999985`}, {4.329999999999998`, 0.04863681249999985`}, {4.349999999999994`, .05163681249999974 '}, {4.3700000000000045`, 0.055636812499999744`}, {4.390000000000001`, 0.06163681249999975`}, {4.409999999999997`, 0.06563681249999975`}, {4.430000000000007`, 0.07063681249999987`}, {4.450000000000003`, 0.07563681249999976`}, {4.469999999999999`, 0.07763681249999976`} , {4.489999999999995`, 0.07763681249999976`}, {4.510000000000005`, 0.07763681249999976`}, {4.530000000000001`, 0.07763681249999976`}, {4.549999999999997`, 0.07463681249999987`}, {4.569999999999993`, 0.07163681249999976`}, {4. 590000000000003`, 0.06863681249999987`}, {4.609999999999999`, 0.06663681249999986`}, {4.6299999999999955`, 0.06363681249999975`}, {4.650000000000006`, 0.06163681249999975`}, {4.670000000000002`, 0.05963681249999975`}, {4.689999999999998`, 0.05963681249999975`}, {4.709999999999994` , 0.05963681249999975`}, {4.730000000000004`, 0.06063681249999986`}, {4.75`, 0.06163681249999975`}, {4.769999999999996`, 0.06363681249999975`}, {4.790000000000006`, 0.0620795088521887`}, {4.810000000000002`, 0.059369054634550764`}, {4.829999999999998`, .053696366699785406 '}, {4.849999999999994`, 0.04688690249267567`}, {4.8700000000000045`, 0.03781840624999988`}, {4.890000000000001`, 0.027168350527011807`}, {4.909999999999997`, 0.01682594903416704`}, {4.930000000000007`, 0.007795621910449408`}, {4.950000000000003`, 0.0019977950011546724`} , {4.969999999999999`, 0.`}};

and i want to use the NFourierTrigSeries to analyze that data and create a variance density spectrum

<< FourierSeries
[CapitalDelta]x = 0.02;
Length1 = 5;
Echos1 = Interpolation[EchoHannatoS1];
Plot[{Echos1[x]}, {x, 0, Length1 - [CapitalDelta]x}]Profilecyclic1[x_] = Piecewise[{{Echos1[x], 0 <x < Length1 - [CapitalDelta]x}}, 0];
Plot[Profilecyclic1[x], {x, 0 - 1, Length1 + 1}]
CyclicFunction1 = Interpolation[Table[{x, Profilecyclic1[x]}, {x, 0, Length1, 0.02}], PeriodicInterpolation -> True];
f0 = 1 / Length1; (* fundamental frequency *)
fc = 1 / [CapitalDelta]x; (* max frequency *)
M = IntegerPart[ fc/(2 f0)] ; (* order of the Fourier series *)
FourierFunction1[x_] =
NFourierTrigSeries[CyclicFunction1[x], x, M, FourierParameters -> {1, 2 [Pi] f0}];
Plot[FourierFunction1[x], {x, 0, Length1}]A0 = FourierFunction1[x][[2, 1]]; (* Mean value of the fourier function *)
AllCoefficients = Reap[For[i=1i[For[i=1i< (M + 1), i++, Sow[i]; Sow[f0 i]; 
   Sow[Coefficient[FourierFunction1[x], Cos[ 2 [Pi] f0 i  x]]]; 
   Sow[Coefficient[FourierFunction1[x], Sin[ 2 [Pi] f0 i  x]]]; 
   Sow[[Sqrt]((Coefficient[FourierFunction1[x], 
         Sin[ 2 [Pi] f0 i  x]])^2 + (Coefficient[
         FourierFunction1[x], Cos[2 [Pi] f0 i  x]])^2)]];] [[2,1]]~Partition~5;
(* i'm creating a table with the values of i, f0i ,Ai, Bi,ai *)
MatrixForm[Prepend[AllCoefficients, {"i", "fi", "Ai", "Bi", "ai"}]];
Dimensions[AllCoefficients];
fi = AllCoefficients[[All, 2]];
Ai = AllCoefficients[[All, 3]];
Bi = AllCoefficients[[All, 4]] ;
ai = AllCoefficients[[All, 5]];
(* verify the Parseval's theorem --> the result has to be approximately 0 *)
ParsevalValue = (f0 *
NIntegrate[(FourierFunction1[(FourierFunction1[x]) ^ 2, {x, 0, Length1}]) - ( one/
2 * (Sum[ai[[i]]^ 2, {i, 1, M}]))
checkfunction1[x_] = Sum[Ai[[i]]* (Cos[two[2[Pi] f0 i x]), {i, 1, M}]+ Sum[Bi[[i]]*(Without[two[2[Pi] f0 i x]), {i, 1, M}];
Plot[checkfunction1[checkfunction1[x], {x, 0, Length1}]Spectrum = Transpose @ {fi, (ai ^ 2) / (2 f0)};
ListLinePlot[{Spectrum}, PlotRange -> All]
S = Interpolation[Spectrum];
LogLogPlot[{S[f]}, {f, f0, M f0}, PlotLegends -> "Expressions", AxesLabel -> {"f", "S (f)"}, PlotRange -> {10 ^ (- 15), 10 ^ (- 3 )}]

The results seem to be correct. I have all that I need for my analysis. However, if I Plot the fourier coefficients I get these results

ListPlot[Ai, PlotRange -> All, Filling -> Axis]
ListPlot[Bi, PlotRange -> All, Filling -> Axis]

and I can not understand why the coefficients Ai and Bi have lost their symmetric and anti-symmetric properties.

Can anybody point out the error that i have done? Maybe something about the Assumption of the FourierTrigSeries or about the FourierParameters?

I'm asking these questions cause I need that Fourier coefficients have symmetric properties in order to continue my analysis.