Pensieve header: The $k=2$ building blocks.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects/SL2Invariant"];
<< SL2Invariant.m
$QZipFail=True;
Block[{$k=2}, atoms={
am->(Subscript[am, i,j->k]),bm->(Subscript[bm, i,j->k]),dm->(Subscript[dm, i,j->k]),R->(Subscript[R, i,j]),(OverBar[R])->(Subscript[(OverBar[R]), i,j]),P->(Subscript[P, i,j]),aS->(Subscript[aS, i]),(OverBar[aS])->(Subscript[(OverBar[aS]), i]),bS->(Subscript[bS, i]),(OverBar[bS])->(Subscript[(OverBar[bS]), i]),dS->(Subscript[dS, i]),aΔ->(Subscript[aΔ, i->j,k]),bΔ->(Subscript[bΔ, i->j,k]),dΔ->(Subscript[dΔ, i->j,k]),C->(Subscript[C, i]),(OverBar[C])->(Subscript[(OverBar[C]), i]),Kink->(Subscript[Kink, i]),(OverBar[Kink])->(Subscript[(OverBar[Kink]), i]),b2t->(Subscript[b2t, i]),t2b->(Subscript[t2b, i])
}] // Column

Print[degrule=Thread[{a,b,α,β,ξ,η,x,y,ℏ,γ,ϵ,t,τ,T,B,𝒜}->{1,1,-1,-1,-1,-1,1,1,-2,1,1,2,-2,0,0,0}]];
atoms /. (Subscript[𝔼, sp__])[L_,Q_,P_]:>(𝔼[L,Q,P]≡(𝔼[L,Q,Normal@P] /. {(Subscript[v_, i_]):>(s^(v/.degrule))(Subscript[v, i]), (v:ℏ|ϵ|γ):>(s^(v/.degrule))v}))

Print[degrule=Thread[{a,b,α,β,ξ,η,x,y,ℏ,γ,ϵ,t,τ,T,B,𝒜}->{0,1,0,-1,0,-1,0,1,-1,0,1,1,-1,0,0,0}]];
atoms /. (Subscript[𝔼, sp__])[L_,Q_,P_]:>(𝔼[L,Q,P]≡(𝔼[L,Q,Normal@P] /. {(Subscript[v_, i_]):>(s^(v/.degrule))(Subscript[v, i]), (v:ℏ|ϵ|γ):>(s^(v/.degrule))v}))

Print[degrule=Thread[{a,b,α,β,ξ,η,x,y,ℏ,γ,ϵ,t,τ,T,B,𝒜}->{1,0,-1,0,-1,0,1,0,-1,1,0,1,-1,0,0,0}]];
atoms /. (Subscript[𝔼, sp__])[L_,Q_,P_]:>(𝔼[L,Q,P]≡(𝔼[L,Q,Normal@P] /. {(Subscript[v_, i_]):>(s^(v/.degrule))(Subscript[v, i]), (v:ℏ|ϵ|γ):>(s^(v/.degrule))v}))

Column[atoms /. (Subscript[𝔼, sp__])[L_,Q_,P_]:>(Subscript[𝔼𝔼, sp])[L,Q,CF@Log[P]]]

degs=atoms /. (Subscript[𝔼, sp__])[L_,Q_,P_]:>Exponent[Normal@CF@Log@P /. {(Subscript[(v:x|y|ξ|η), i]):>λ (Subscript[v, i]),(Subscript[(v:a|β), i]):>(λ^2) (Subscript[v, i])},λ]

Last/@degs

Max[Last/@degs]