We learn from Moshe Rabbeinu the proper method of asking for forgiveness. We saw that he was successful on achieving atonement for the sin of the Golden Calf.
First, Moshe mentioned זכות אבות, the merit of our fathers. In essence, he was telling G-d not to forgive us for our sake, but for the sake of Avraham, Yitzchak, and Yakov.
After this, Moshe reminded Hashem that if He punished the Jewish people in the desert, He would look weak. Why should He allow the Egyptians to be mocking, and questioning of G-d’s ultimate power. This would be a desecration of His holy Name.
And finally, Moshe used the formula taught to him by G-d, how to change harsh judgement into mercy. This would come about through the utterance of the Thirteen Attributes of Hashem. We include this in our Selichot prayers, and say them at least seven days during Ne’ila, at the end of Yom Kippur.
This is the formula for seeking Hashem’s favor. Do it for the sake of the Avot, and His holy Name. And be merciful through the utterance of the Thirteen Attributes.