Brandt matrix

In mathematics, Brandt matrices are matrices, introduced by , that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra.
 calculated the traces of the Brandt matrices.
Let "O" be an order in a quaternion algebra with class number "H", and "I"i...,"I""H" invertible left "O"-ideals representing the classes. Fix an integer "m". Let "e""j" denote the number of units in the right order of "I""j" and let "B""ij" denote the number of α in "I""j"−1"I""i" with reduced norm N(α) equal to "m"N("I""i")/N("I""j"). The Brandt matrix "B"("m") is the "H"×"H" matrix with entries "B""ij". Up to conjugation by a permutation matrix it is independent of the choice of representatives "I""j"; it is dependent only on the level of the order "O".