using "at your convenience" while asking a question?

I am writing an email for a professor in a university and asking some question about a program. First, I asked the questions and in the ending paragraph I wrote:

I would be grateful if you answer my questions at your best convenience.

Does it elevate the overall politeness of the mail or it is incorrect?

Solution 1:

As in:

I would be grateful for a reply at your convenience.

at (one's) convenience TFD an idiom

• When one has the time to do something.
• eg. No rush, you can fill out that paperwork at your convenience.

You have posed the questions in the body of the letter. No need to re-iterate the request to answer them. And 'best' can be pushy ... use it if you must.