The Goths in the time of Constantine were heavily Romanised. The northern border of the Roman Empire on the Danube was quiet and peaceful. The Goths traded with the Romans and supplied mercenaries for the Roman Army. The Roman Empire by this time was Christian though the Goths were not. They still worshiped Pagan gods; however that was changing. Between the Goths capturing Christian Romans in prior warfare from the third century and integrating them into Gothic culture as slaves and peasants, and the increasing Romanisation of the Goths, Christianity began to be adopted.

Emperor Constantine from SA Moberly's photostream.

Constantine moved the center of the Roman Empire to Constantinople and established regular contact with the Goths. With this came Greek speaking, as opposed to Latin speaking, missionaries. Additionally Goths learnt Greek and studied Christianity in the Roman Empire.

Apparently there was one such student named Ulfia who translated the Greek bible to the Goth language; part of which still survives today. An innovation to achieve that was to create an alphabet for the Gothic language. Quite remarkable.

There was also philosophical disagreement over Christianity. Alessandro Barbero writes:

[C]hristians of that time were not yet in agreement about the definition of the Trinity, about Christ's nature, or about the relationship between the Father and the Son, and the faith was split into ferociously quarreling factions.

One of which was the Catholics, or universalists. It does point to the Goths being far from the barbarian horde of common historical mythology. They were heavily Romanised and culturally were integrating into the wider Roman Empire and culture even if they did not kneel before Constantine as subjects.
Cam Riley: South Sea Republic. Freedom, liberty, equity and an Australian Republic.