## 2 Results
### Superconducting Pairing: Nodal Character and Dimensionality  
The superconducting pairing susceptibility \(\chi^{p-p}\) is computed by dressing the non-local pairing polarisation bubble \(\chi^{0,p-p}(k,iv)\) with the pairing vertex \(\Gamma^{irr,p-p}\) using the Bethe-Salpeter equation in the particle-particle channel. \[\chi^{p-p}=\chi^{0,p-p}\cdot[\mathbf{1}+\Gamma^{irr,p-p}\cdot\chi^{0,p-p}]^{-1} \tag{2}\]  
\(\Gamma^{irr,p-p}\) in the singlet (s) and triplet (t) channels are obtained from the magnetic (spin) and density (charge) particle-hole reducible vertices by:  
\[ \begin{array}{c}\Gamma^{irr,p-p,s}_{\begin{subarray}{c}\alpha_{2},\alpha_{4} \\ \alpha_{1},\alpha_{3}\end{subarray}}(\mathbf{k},i\
u,\mathbf{k}^{\prime},i\
u^{ \prime})=\Gamma^{f-irr}_{\begin{subarray}{c}\alpha_{2},\alpha_{4}\\ \alpha_{1},\alpha_{3}\end{subarray}}(iv,i\
u^{\prime})\\ +\frac{1}{2}[\frac{3}{2}\widetilde{\Gamma}^{p-h_{i},(m)}\\ -\frac{1}{2}\widetilde{\Gamma}^{p-h_{i},(d)}]^{\begin{subarray}{c}\alpha_{2}, \alpha_{3}\\ \alpha_{1},\alpha_{4}\end{subarray}}(iv,-iv^{\prime})_{\mathbf{k}^{\prime}- \mathbf{k},i\
u^{\prime}-iv}\\ +\frac{1}{2}[\frac{3}{2}\widetilde{\Gamma}^{p-h_{i},(m)}\\ -\frac{1}{2}\widetilde{\Gamma}^{p-h_{i},(d)}]^{\begin{subarray}{c}\alpha_{4}, \alpha_{3}\\ \alpha_{1},\alpha_{2}\end{subarray}}_{\alpha_{1},\alpha_{2}}(iv,i\
u^{\prime}) _{-\mathbf{k}^{\prime}-\mathbf{k},-i\
u^{\prime}-iv}\end{array}  \tag{3}\]  
Figure 4: Strain and temperature dependence of susceptibilities: (**a–e**) Imaginary part of the dynamic spin susceptibility \(\chi(\mathbf{q},\omega)\) at some high symmetry points in the Brillouin zone for different strains \(\epsilon_{x}\). The unstrained compound shows a spin fluctuation spectrum strongly peaked at \((0.3,0.3,0)\). With increasing strain, fluctuations become more strongly peaked at \((0.3,0.3,0)\), while they get suppressed at the ferromagnetic vector and remain fully gapped at the anti-ferromagnetic vector. (**d**) We also observe an increment in spin susceptibility at a vector \((0.5,0.25,0)\) under strain. (**f**) With lowering temperature, the IC peak at \(\mathrm{q}=(0.3,0.3,0)\) starts to diverge for strains \(\epsilon_{x}>0.6\%\), signalling an instability towards a spin density wave order. \[\Gamma^{irr,p-p,t}_{ \begin{subarray}{c}a_{2},a_{4}\\ a_{1},a_{3}\end{subarray} }(\mathbf{k},i\
u,\mathbf{k}^{\prime},i\
u^{\prime})= \Gamma^{f-irr}_{ \begin{subarray}{c}a_{2},a_{4}\\ a_{1},a_{3}\end{subarray} }(iv,iv^{\prime})\] \[-\tfrac{1}{2}\{\tfrac{1}{2}\widetilde{\Gamma}^{p-h,(t)}\}_{ \begin{subarray}{c}i_{1},a_{2},a_{3}\end{subarray} }(iv,-iv^{\prime})_{ \mathbf{k}^{\prime}-\mathbf{k},i\
u^{\prime}-iv}\] \[+\tfrac{1}{2}[\widetilde{\Gamma}^{p-h,(t)}]_{ \begin{subarray}{c} i_{1},a_{2},a_{3}\end{subarray} }(iv,-iv^{\prime})_{\mathbf{k}^{\prime}- \mathbf{k},i\
u^{\prime}-iv}\] \[+\tfrac{1}{2}[\widetilde{\Gamma}^{p-h,(t)}]_{ \begin{subarray}{c} i_{1},a_{2},a_{3}\end{subarray} }(iv,i\
u^{\prime})_{-\mathbf{k}^{\prime}- \mathbf{k},i\
u^{\prime}-iv} \tag{4}\]  
Finally, \(\chi^{p-p}\) can be represented in terms of the eigenvalues \(\lambda\) and eigenfunctions \(\phi^{\lambda}\) of the Hermitian particle-particle pairing matrix. \[\chi^{p-p}(k,k^{\prime}) = \sum_{\lambda}\frac{1}{1-\lambda}\cdot(\sqrt{\chi^{0,p-p}(k)} \cdot\phi^{\lambda}(k)) \tag{5}\] \[\cdot( \sqrt{\chi^{0,p-p}(k^{\prime})}\cdot\phi^{\lambda}(k^{\prime}))\]  
The pairing susceptibility diverges when the leading eigenvalue \(\lambda\) becomes unity. The corresponding eigenfunction represents the momentum structure of \(\chi^{p-p}\). Unconventional superconductivity in SRO is multi-orbital in nature with multiple competing instabilities. In our previous work [17], we performed a thorough analysis of all possible singlet and triplet instabilities in SRO and associated with a particular symmetry group.