$\begin{array}{l} a = \sqrt[3]{{\dfrac{{25 + \sqrt {621} }}{2}}} + \sqrt[3]{{\dfrac{{25 - \sqrt {621} }}{2}}}\\ {a^3} = 25 + 3\sqrt[3]{{\dfrac{{\left( {25 + \sqrt {621} } \right)\left( {25 - \sqrt {621} } \right)}}{4}}}a\\ \Rightarrow {a^3} = 25 + 3a \Rightarrow {a^3} - 3a - 25 = 0\\ \Rightarrow y = \dfrac{1}{3}\left( {1 - a} \right) = \dfrac{1}{3} - \dfrac{1}{3}a\\ T = 2{y^3} - 2{y^2} + 2020\\ T = 2.{\left( {\dfrac{1}{3} - \dfrac{a}{3}} \right)^3} - 2{\left( {\dfrac{1}{3} - \dfrac{a}{3}} \right)^2} + 2020\\ T = 2\left( {\dfrac{1}{{27}} - 3.\dfrac{1}{9}.\dfrac{a}{3} + 3.\dfrac{1}{3}.\dfrac{{{a^2}}}{9} - \dfrac{{{a^3}}}{{27}}} \right) - 2\left( {\dfrac{1}{9} - \dfrac{{2a}}{9} + \dfrac{{{a^2}}}{9}} \right) + 2020\\ T = \dfrac{2}{{27}} - \dfrac{{2a}}{9} + \dfrac{{2{a^2}}}{9} - \dfrac{{2{a^3}}}{{27}} - \dfrac{2}{9} + \dfrac{{4a}}{9} - \dfrac{{2{a^2}}}{9} + 2020\\ T = - \dfrac{4}{{27}} + \dfrac{{2a}}{9} - \dfrac{{2{a^3}}}{{27}} + 2020\\ T = - \dfrac{4}{{27}} + \dfrac{{2a}}{9} - \dfrac{{2\left( {3a + 25} \right)}}{{27}} + 2020\\ T = - \dfrac{4}{{27}} + \dfrac{{2a}}{9} - \dfrac{{6a}}{{27}} - \dfrac{{50}}{{27}} + 2020\\ T = - \dfrac{4}{{27}} + \dfrac{{2a}}{9} - \dfrac{{2a}}{9} - \dfrac{{50}}{{27}} + 2020\\ T = - \dfrac{4}{{27}} - \dfrac{{50}}{{27}} + 2020 = 2020 - 2 = 2018 \end{array}$
