Frequency mixer shows 2 FFT spikes

I'm attempting to use a frequency mixer to shift one frequency range to another. Right now, I have a complex sine wave at $$43\ \rm kHz$$. My imaginary value on the complex object is 0.

If I output a frequency-amplitude spectrum it looks like this: Now let's say I want to move that frequency to $$65\ \rm kHz$$. For that I realized that I need to create a new complex sine wave at that frequency and multiply it by my original signal in time.

The problem is, it's not working. I'm getting 2 spikes for some reason. I don't understand why that happens. Here is my code:

typedef std::complex<float> Complex;

Complex chunk[N];
float Fs = 176400; // How many time points are needed i,e., Sampling Frequency
const double  T = 1 / Fs; // At what intervals time points are sampled
float value;
float value2;
for (int i = 0; i < N; i++)
{
value2 = (float)(1 * sin(2 * M_PI * 43000 * (i * T))); // Original Signal
Complex value3 = {(float)(1 * sin(2 * M_PI * 12000 * (i * T))), 0}; // The frequency I want to add
double multiplier = 0.5 * (1 - cos(2*M_PI*i/256)); // Hamming Window
chunk[i] = {value2 * multiplier, 0 };// generate (complex) sine waveform
chunk[i] = chunk[i] * value3; // Frequency Mixer
}