Dimension1 Tetrahedral
Group $G=A_4$ does not have 2-dimensional representation, its binary-extension $SL(2,3)$ has 3 2-dimensional representation, one representation is as follows:
\[\begin{align*} G=\langle \begin{pmatrix} \zeta_4 & 0\\ 0 & \zeta_4^{-1} \end{pmatrix}, \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \begin{pmatrix} (1+\zeta_4)/2 & (-1+\zeta_4)/2\\ (1+\zeta_4)/2 & (1-\zeta_4)/2 \end{pmatrix} \rangle. \end{align*}\]where $\zeta_4$ is a 4-th root of unity.
The Magma code is as follows:
FScale:=CyclotomicField(24:Sparse:=true); F:=FScale; Z4:=RootOfUnity(4); G:=MatrixGroup<2,F| [Z4,0, 0,Z4^(-1)], [0,1, -1,0], [(1+Z4)/2,(-1+Z4)/2, (1+Z4)/2,(1-Z4)/2]>;
The Burnside symbol is
\[\begin{align*} [\mathbb{P}^1 \circlearrowleft G]=&(1,G \circlearrowright k(x),())\\ &+(C_2,1 \circlearrowright k,(1))\\ &+(C_3,1 \circlearrowright k,(1))+(C_3,1 \circlearrowright k,(2)) \end{align*}\]where
\[\begin{align*} C_2=\langle \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \rangle, C_3=\langle \begin{pmatrix} (1+\zeta_4)/2 & (-1+\zeta_4)/2\\ (1+\zeta_4)/2 & (1-\zeta_4)/2 \end{pmatrix} \rangle \end{align*}\]The other 2 projective linear representation of $A_4$ gives the same Burnside symbols.