FIR Filter - Low Pass Filter coefficients to High Pass filter coefficients I need to convert this 12 Tap Low Pass Filter to a High Pass Filter.

The coefficients of the Low Pass Filter are as follows:

h[k] = [0.022 -0.083 -0.083 0.083 0.311 0.417 0.311 0.083 -0.083 -0.083 0.022 0.083]


My understanding is that if its an even-tap - We Multiply the coeffs by -1.

However the answer is:

hT[k] = [-0.022 -0.083 0.083 -0.083 -0.083 0.311 -0.417 0.311 -0.083 0.083 -0.083 -0.022 0.083]


I do not know how this came about. I looked for answers online alot but did not get the answer. Any help is appreciated
HINT:

It looks like they shifted the frequency response by half the sampling frequency, i.e.,

$$H_{HP}\left(e^{j\omega}\right)=H_{LP}\big(e^{j(\omega-\pi)}\big)\tag{1}$$

Frequency shifting corresponds to modulation (multiplication) in the time domain. Now you just have to figure out the modulation sequence that achieves the correct frequency shift.