I believe Faith in Jesus also mean love Jesus, because faith mean trust or believe in jesus teaching and the main teaching of jesus is love.
So It is imposible have a faith in Jesus but hate Jesus.
How about murder or envy is that mean hate Jesus?
Murder mean hate other and John say
1 John 4
20 Whoever claims to love God yet hates a brother or sister is a liar. For whoever does not love their brother and sister, whom they have seen, cannot love God, whom they have not seen.
So base oN that verse murder is hate Jesus, or do not have faith or not save.
That doesn't mean after Being Christian never commit sin, but we have to repent or go to hell.
Now is faith always produce good work
A. We learn Love Jesus mean love other, 1 John 4:20, that may mean helping other in need, It may mean work, helping poor and elder to fix her leaving roof because she don't have money to pay handyman but It is not always, we may love her but we may don't know to fix the roof and don't have money to help her pay handyman.
Faith = love, It may mean work but not always
But
Faith not hate.
Eg murder, envy and other sin.
I don't believe because we are Christian, murder No longer mean hell.
Jesus pay our sin If we have faith in Him.
Remember
Faith mean love.
Repent from our daily sin is not only keep our reward but keep our salvation.
Faith = love
Love may produce good work, may not but love not produce bad work like envy etc.
For loving person that poor and sick, love may not produce work or money, but as long as this person have faith, he save.
For normal, healthy and wealthy person, I believe faith mean good work