How much weight do Math PhD programs place on the subject GRE? I am a master's student in mathematics at a relatively unknown university, studying Number Theory. I have completed readings for my master's thesis and will present this in Spring, along with other readings with another advisor. I have 4.0 gpa throughout my time here, and I believe strong recommendation letters. My research goals are pretty focused, and this will show through on my application (probably mostly so in my personal statement, but also in one rec letter). 

The only thing that is holding me back is this darn math subject GRE test. I have been so focused on research that I did not study for it! It was very fast, with lots of tricky calculus (the "advanced" topics questions are not actually that hard). I have not received my score, but I believe it will be bad, maybe at best 40 percentile, but I'm just guessing. 

So my question: How much weight does math GRE subject test have on my application? Specifically, I'd like to get advice on this from people in pure math on admissions committees (or used to be), since this obviously depends on the university. For instance, how much does it matter for "tier 1", "tier 2", etc. universities (whatever this means)?

I have tried to look up other information on this but little is out there, with the exception of a couple of threads here and there that are not super helpful.

The main reason I ask is that I am spending potentially a lot of money on applications, most of which require this tricky calculus test! But I'd rather not apply and save the money if it won't matter.
I worked in the admissions office of the graduate school of arts and sciences at Harvard University.

While the GRE is important, there are a kaleidoscope of other factors that admissions takes into consideration.

You stated, quite eloquently, what happened, which was that you were so engaged in research that it hurt your study time for the GRE.

Admission officers value applicants who display intellectual curiosity but also honesty.

In your statement of purpose, explain what happened

If you were wanting to get into a top-five university, what about applying as a special student. The GRE is not required, but I must warn you, it is still as competitive as applying as a degree seeking student.

Just a thought, and wishing you all the best in your educational pursuits.