What is the probability distribution of the DFT of a real Gaussian white noise random vector?

import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

N = 1024
num_samples = 100000
X = np.random.normal(size=(num_samples, 1024))
X_hat = np.fft.fft(X, axis=-1)

fig, ax = plt.subplots(nrows=2, ncols=2, figsize=(10, 8))
fig.suptitle('Distribution of DFT of real-valued Gaussian white noise\n$N=1024$, # Samples = 100,000')
ax[0,0].grid()
ax[0,0].hist(np.real(X_hat[:, 0]), bins=100, density=True, label='Observed')
dom = np.linspace(-140, 140, 100)
ax[0,0].plot(dom, sp.stats.norm.pdf(dom, scale=np.sqrt(N)), label='Theoretical')
ax[0,0].legend()
ax[0,0].set_title('Distribution of $\\hat{X}_0^\\mathrm{real}$')
ax[0,1].grid()
ax[0,1].hist(np.imag(X_hat[:, 0]), bins=100, density=True, label='Observed')
dom = np.linspace(-140, 140, 100)
ax[0,1].set_title('Distribution of $\\hat{X}_0^\\mathrm{imag}$')
ax[1,0].grid()
ax[1,0].hist(np.real(X_hat[:, 3]), bins=100, density=True, label='Observed')
dom = np.linspace(-140, 140, 100)
ax[1,0].plot(dom, sp.stats.norm.pdf(dom, scale=np.sqrt(N/2)), label='Theoretical')
ax[1,0].set_title('Distribution of $\\hat{X}_3^\\mathrm{real}$')
ax[1,1].grid()
ax[1,1].hist(np.imag(X_hat[:, 3]), bins=100, density=True, label='Observed')
dom = np.linspace(-140, 140, 100)
ax[1,1].plot(dom, sp.stats.norm.pdf(dom, scale=np.sqrt(N/2)), label='Theoretical')
ax[1,1].set_title('Distribution of $\\hat{X}_3^\\mathrm{imag}$')

plt.show()