After he flunks his trigonometry test, what conclusion does Gene come to regarding himself and Finn? How does the idea of hate fir into his thoughts?

"Gene, an academic perfectionist, laments his poor performance on the test to Finny; Finny mocks Gene's ambition to be first in their class, and Gene begins to believe that Finny doesn't want him to do well in school, so that he will come out ahead. Finny excels in athletics, and is definitely the best in the school; Gene knows that he can be the best in the school in academics, but thinks that Finny's high-jinks and his attempts to take up Gene's time are Finny's attempts to make sure that he comes out ahead in the relationship. Gene's jealousy, whether merited or not, begins to take him over; he decides that he cannot trust his "friend," and Finny's statement, that had been so touching the previous evening, of Gene being his best friend, Gene now believes to be false."

Hatred comes into play with the rivalry Gene feels with Finny. He believes that Finny hates his academic success as much as he hates Finny's athletic abilities.