How to encrypt a string using public key cryptography

I am trying to implement my own RSA encryption engine. Given these RSA algorithm values:

p = 61. // A prime number.
q = 53. // Also a prime number.
n = 3233. // p * q.
totient = 3120. // (p - 1) * (q - 1)
e = 991. // Co-prime to the totient (co-prime to 3120).
d = 1231. // d * e = 1219921, which is equal to the relation where 1 + k * totient = 1219921 when k = 391.

I am trying to write a method to encrypt each byte in a string and return back an encrypted string:

public string Encrypt(string m, Encoding encoding)
{
    byte[] bytes = encoding.GetBytes(m);
    for (int i = 0; i < bytes.Length; i++)
    {
        bytes[i] = (byte)BigInteger.ModPow(bytes[i], e, n);
    }
    string encryptedString = encoding.GetString(bytes);
    Console.WriteLine("Encrypted {0} as {1}.", m, encryptedString);
    return encryptedString;
}

The obvious issue here is that BigInteger.ModPow(bytes[i], e, n) may be too large to fit into a byte-space; it could result in values over 8 bits in size. How do you get around this issue while still being able to decrypt an encrypted string of bytes back into a regular string?

Update: Even encrypting from byte[] to byte[], you reach a case where encrypting that byte using the RSA algorithm goes beyond the size limit of a byte:

public byte[] Encrypt(string m, Encoding encoding)
{
    byte[] bytes = encoding.GetBytes(m);
    for (int i = 0; i < bytes.Length; i++)
    {
        bytes[i] = (byte)BigInteger.ModPow(bytes[i], e, n);
    }
    return bytes;
}

Update: My issue is that encryption would cause a greater number of bytes than the initial input string had:

public byte[] Encrypt(string m, Encoding encoding)
{
    byte[] bytes = encoding.GetBytes(m);
    byte[] returnBytes = new byte[0];
    for (int i = 0; i < bytes.Length; i++)
    {
        byte[] result = BigInteger.ModPow(bytes[i], (BigInteger)e, n).ToByteArray();
        int preSize = returnBytes.Length;
        Array.Resize(ref returnBytes, returnBytes.Length + result.Length);
        result.CopyTo(returnBytes, preSize);
    }
    return returnBytes;
}

public string Decrypt(byte[] c, Encoding encoding)
{
    byte[] returnBytes = new byte[0];
    for (int i = 0; i < c.Length; i++)
    {
        byte[] result = BigInteger.ModPow(c[i], d, n).ToByteArray();
        int preSize = returnBytes.Length;
        Array.Resize(ref returnBytes, returnBytes.Length + result.Length);
        result.CopyTo(returnBytes, preSize);
    }
    string decryptedString = encoding.GetString(returnBytes);
    return decryptedString;
}

If you ran this code like this:

byte[] encryptedBytes = engine.Encrypt("Hello, world.", Encoding.UTF8);
Console.WriteLine(engine.Decrypt(encryptedBytes, Encoding.UTF8));

The output would be this:

?♥D
?♥→☻►♦→☻►♦oD♦8? ?♠oj?♠→☻►♦;♂?♠♂♠?♠

Obviously, the output is not the original string because I can't just try decrypting each byte at a time, since sometimes two or more bytes of the cypher-text represent the value of one integer that I need to decrypt back to one byte of the original string...so I want to know what the standard mechanism for handling this is.