Your conversion is considered income for many purposes -- not just the regular tax purposes (if you convert $40,000, it's $40,000 additional taxable income as I'm sure you are aware of).
But also it raises your AGI by $40,000 and that raises your "stealth taxes" which are all dependent on the AGI level,
1. The amount you pay on your Social Security benefits will be higher, unless you are already maxed out at the 85% level.
2. Your Medicare Part B and Part D premiums may be higher -- Google "medicare irmaa"
3. If you were in the ACA, it would affect your ACA premium subsidy or get rid of it entirely
These were all considerations I had to take into account in recent years when deciding on my Roth conversion amount (in three recent years I converted $0 or a piddling amount because of the above)
I don't think the Roth conversion calculators out there consider these elements, but it's real money.
I am so so glad that I've been converting pieces of my traditional IRA almost every year starting in 1999. Had I not, I would have been facing enormous RMD's soon (which of course also raise one's taxable income and one's AGI and therefore also one's stealth taxes).
P.P.S. - if it's big, it could screw you up as far as estimated tax payments and penalties. For example, I do my IRA RMD's (on my beneficiary IRA) and my Roth conversion in December. I pay more estimated tax January 15 to cover that (what you pay on January 15 covers the October 1 - December 31 quarter). That should work out but TurboTax complains and comes up with a penalty -- thinking this "income" is spread out evenly over the year, but I'm paying estimated tax on most of it in the 4th quarter. (IRA RMDs and Roth conversions are "income" )
One can "annualize" by filling out a special tax form 2210, where you list your income by quarter as well as your estimated tax payments by quarter, and escape the penalty, but then my understanding is that TurboTax won't let you electronically file, GRRRR. I only had a small conversion in 2019, and owed a $1 penalty on my federal, and chose not to annualize because it wasn't worth the work for $1, and I'd have to snail-mail it in.
I owed Minnesota a $2 penalty for the same reason, and again I chose not to annualize. But TurboTax required that for Minnesota, any penalty means I can't electronically file. (And although I could annualize away the penalty, TT says I can't electronically file if i annualize). So I had to snail mail it in.