You believe wrong. Antisemitism, even in Rome, well predates Christianity. I suggest you read Constantine's Sword by James Carroll, a very well-researched look into the history of European antisemitism.
Roman antisemitism stems from several issues:
1. Most people Romans conquered melded Roman gods with their own pantheon. Jews did not -- this was something of an affront.
2. In Rome, alliances were usually forged over dinner. Jewish laws of kashrut prohibited Jews from eating in Roman homes. This was seen as alien and inhospitable.
3. As set forth before, the Emperor who first conquered Jerusalem around 63 B.C. His dynasty's legitimacy was based in part on his conquest of the region.
4. The Sicarii, Zealots and other Jewish rebels made this conquest infirm (which is why the charge that Jesus was being treasonous to Rome by calling himself King of Jews was dealt with so harshly). Rome was brutal to Jews. by some historian's counts, one in three Jews worldwide were killed during Rome's pacification of Jerusalem. That equals -- proportionally -- the proportion of the world's Jews killed in the Holocaust. All of this occurred between 106 B.C. and 135 CE, well before Christianity had a foothold in Rome.
When Christianity did begin to take hold, it went to great lengths to divorce itself from Judaism entirely. Partly, this was not because Christianity is inherently antisemitic, but because it was trying to convert a people who were already antisemitic. So it adopted pre-existing Roman antisemitism and incorporated it.