With 13 HCP you should open the bidding. If you have 5 cards in a major suit you want to open 1 of that major.
Therefore to open the bidding 1♥ you need 13 HCP and 5♥. That's it.
When partner opens 1♥ we no longer need 13 points to make a bid. Since partner opened the bidding they have the opening hand. We just need 6 HCP to respond. With 6 HCP we need to respond. To bid a new suit we just need 4 cards in that suit. To support partner's ♥ suit we should have 3 cards in the suit. (Partner needs 5 cards in the suit to open remember. So to get an 8 card fit we only need 3.)
If partner opens 1♥ and you have a fit you should probably tell partner about it right away.
The only bids that limit your hand are 1NT and 2♥. Both these bids limit your hand to 6 to 10 points and partner can pass both those bids.
If you respond with a new suit at the 1 level you might only have 6 points, but you could have 13 points -- or even 20 points. Partner MUST bid again. But we can let partner worry about that.
If you respond with a new suit at the 2 level you must have at least 10 HCP. Therefore sometimes you might have a hand with ♣ or ♦ that you can't tell partner about. You might get stuck bidding 1NT with an unbalanced hand or you might have to bid a shorter major suit rather than a longer minor suit. Such is life.
Therefore if your partner opens the bidding 1♥ here is a table that states your only possible responses if you have 6 to 9 points:
Bid
Points
Number in suit
1♠
6 or more points and
at least 4 in the suit
1NT
6 to 10 points and
No 4 card ♠ suit and no 3 card ♥ suit
2♥
6 to 10 points and
at least 3 cards in the ♥ suit
With that in mind what do you respond when your partner opens 1♥?
Hand 1
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