We don't know the secular historical crime that Yeshua ben Yosef was executed for. The reason we don't know this is that there are no such records from Roman times. All we have are texts written a lot later, by people who was not there and never met this Yeshua, and was only told about it.
In fact, most texts that we have are written by people, who got all their information from Paul of Tarsus, who was not there and never met Yeshua ben Yosef himself. As such, the information we have is at best third-hand. In addition to this, it's generally written with specific aims in mind.
As such, one of the text claims that Pontios Pilatos said "I find no fault in this man" with regards to Yeshua. But do we know that for sure? No, this is likely to be a part of an effort of the Romans who wrote the texts to blame the Jews for the Crucifixion, and putting less blame on the Romans, in an effort to make the religion more palatable to Romans.
But it was a crucifixion which was a roman punishment, and one especially used for lower classes. The Jewish bible does not prescribe crucifixion as a punishment at all. If Yeshua's crime had been a Jewish religious affair, he would rather have been stoned (or possibly strangulated, if he was deemed to prophesy).
As such, there are simply no reliable sources on this, and therefore we do not know the reason he was executed. Any discussion about it is therefore not much than at best educated speculation.
Many of the crimes that would have been punishable by execution would have been incompatible with the Jewish faith anyway. Tradition has it that three men was crucified that day, and that the other two were thieves. If Yeshua ben Yosef was executed for crimes like this his followers would have kept quiet about it and in that case they would have made up an acceptable reason. But yet again, we don't know if the story of these two thieves is true, it may very well also have been made up. It's very convenient from a religious standpoint, as one of the thieves is regretful and the other not, as this forms a good basis for a moral story, which is how it's used in the bible.
A probably more likely, and for his followers, acceptable, reason would be if he was executed for sedition or revolting against the Roman rule. This is also compatible with him being a religious leader, as the Jewish religious leadership was in power with the support of the Roman rulers. As such it is possible that if he was deemed to try to overthrow the Jewish religious leadership, he might have been sentenced as trying to overthrow the Roman power in Jerusalem, and that would definitely have been punishable by crucifixion, even for a high-standing citizen.
Again, if this was the case, the later Roman authors of the biblical texts would have kept quiet about that, as they had no interest in portraying their religion as opposed to Roman rule. As such this is a quite popular theory, but again it's just speculation.