Sign with the private exponent and verify with the public one, on a number or on a SHA-256 hash, with the textbook pitfalls explained.
Procedure
How it works
An RSA signature uses the key pair the other way round: the signer computes $s = h^d \bmod N$ with the private exponent, and anyone can check $s^e \equiv h \pmod N$ with the public key $(N, e)$. Only the holder of $d$ can produce an $s$ that passes, as long as taking $e$-th roots modulo $N$ is hard.
Signing the message itself as a number is insecure. Anyone can choose $s$ and publish the pair $(s^e \bmod N, s)$, a valid signature on a message they do not control (an existential forgery), and two signatures multiply into a signature on the product of the messages. That is why $h$ is a hash of the message.
Real schemes go further and pad the hash before exponentiation: RSA-PSS (randomized, with a security proof in the random oracle model) or the older PKCS #1 v1.5. In this tool the SHA-256 digest is reduced modulo the toy $N$ so that it fits; with a 2048-bit key the padded hash fits without reduction.
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