Conviction deals with the sin of unbelief (John 16:7-11). The Holy Spirit convicts and convinces us of our need to believe in Christ. Concerning the Holy Spirit, Jesus said, "He will convict the world of sin, and of righteousness, and of judgment: of sin, because they do not believe in me" (John 16:8-9).
Conviction prepares us for repentance toward God and faith toward our Lord Jesus Christ (Acts 20:21). God's grace enables us to repent and believe the gospel of Christ. God's goodness leads us to repent (Romans 2:4).
Conviction comes before we repent and believe on the Lord Jesus Christ. Repentance is a change of mind through the renewing of the Holy Spirit transforming our lives (Romans 12:2). Faith is an abiding gift of the Holy Spirit. God's preceding grace enables us to repent and believe on the Lord Jesus Christ (Acts 18:27).
Conviction and new birth are both the work of the Holy Spirit. Whoever believes that Jesus is the Christ is born of God, and everyone who loves Him who begot also loves him who is begotten of Him (1 John 5:1). Both faith to believe and love are gifts of the Holy Spirit (1 Corinthians 13:13). From conviction to new birth, it's the Holy Spirit's work within us.