No, A harmonic major and E harmonic major are too different scales. E harmonic major is not a mode of A harmonic major. See here are the notes that consist in those 2.

A harmonic major: A, B, C#, D, E, F, G#
E harmonic major: E, F#, G#, A, B, C, D#

See how they are different. Now you can play the A harmonic major scale, starting with E. That's playing the 5th mode of the A harmonic major.

The notes would be: E, F, G#, A, B, C#, D.

As you can see the same notes as the A harmonic major, just starting with E. So yeah, that's how a mode is constructed, but it's function is different from the Root scale. The 5th mode (E) of the A harmonic minor would work well over a E7b9 chord, when the A harmonic major would NOT work over a A7b9. As you can see the root is the key. The reason is the clashing of 9th's between scale and chord. The 9th of a A harmonic major is B, when the chord A7b9, the 9th is Bb. Also, the 7th's are clashing too. When you look at the 5th mode (E) of the A harmonic minor, it works because the notes of the chord and scale are the same. A E7b9 chord is made up of E, G#, B, D, and F. All are found in the mode, so it works. Each scale/mode works with one or more chords, modes are very useful for these reasons.