I come here the first time.句子成分问题

I come here the first time.句子成分问题
我知道the first time是作状语,那它是作什么状语呢?时间状语吗?如果是作时间状语的话,那换一种说法：This is the first time I come here.很明显This is the first time是主句,I come here是从句,并且从句前面省略了一个连接词.如果是时间状语的话,连接词就应该是when,但是This is the first time when I come here.读起来总觉得怪怪的.感觉应该是This is the first time that I come here.其中that引导the first time的同位语从句.
绕来绕去可能把大家绕糊涂了,我再说一遍我的问题.我的问题是the first time在I come here the first time这句话里作的是什么状语?