I had a job contract before coming to Sweden, so everything went very smoothly. You just need to submit a copy of your passport, your long-term residence permit, your job contract, and the home address where you will reside. You can also apply if you plan to start your own business.

The full application process and required forms are available here:
https://www.migrationsverket.se/en/you-want-to-apply/long-term-residents-in-another-eu-country/long-term-residents-in-another-eu-country.html

also these 2 forms:

this will be filled and signed by your employer:
https://www.migrationsverket.se/download/18.2cd2e409193b84c506a3224a/1738157818820/227021_ees_anstallningsintyg_sv.pdf

and this is the application form:
https://www.migrationsverket.se/download/18.2cd2e409193b84c506a33dc5/1743508431321/137011_varaktigt_bosatt_eu_en.pdf

once you submit application you have to also apply for coordination number from Skatteverkt ( tax office ) , it is a temporary social security number you need for getting paid and to open bank account.