As far as I know it's not a "waiver" in the sense that you have to ask or apply for it.
Pub 1060 states:
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Newly Incorporated or Qualified Corporation Exception
Newly incorporated or qualified corporations are not required to pay the minimum franchise tax in their first
taxable year.
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I would think that the TTBusiness California program should "know" this and handle it correctly.
I'm not sure what you mean by " it was my first year of doing business by a few weeks." It seems like it is, or it isn't.
