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We study the representation theory of the Soergel calculus algebra $ A_w :=\mbox{End}_{{\mathcal D}_{(W,S)}} (\underline{w}) $ over $\mathbb C$ in type$\tilde{A}_1$. We generalize the recent isomorphism between the nil-blobalgebra ${\mathbb{NB}}_n$ and $ A_w $ to deal with the two-parameter blobalgebra. Under this generalization, the two parameters correspond to the twosimple roots for $\tilde{A}_1$. Using this, together with calculationsinvolving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of $\mathbb{NB}_n$, we obtain a concrete diagonalization of the matrix of thebilinear form on the cell module $\Delta_w(v) $ for $ A_w $. The entries of thediagonalized matrices turn out to be products of roots for $\tilde{A}_1$. Weuse this to study Jantzen type filtrations of $ \Delta_w(v)$ for $A_w $. Weshow that at enriched Grothendieck group level the corresponding sum formulahas terms $ \Delta_w(s_{\alpha }v)[ l(s_{\alpha }v)- l(v)] $, where $[ \cdot ]$ denotes grading shift.

Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra