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Zagier's well‐known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight    1  2    to traces of values of the modular   j  ‐function at imaginary quadratic points. A real quadratic analogue was recently studied by Duke, Imamoḡlu, and Tóth. They showed that the coefficients of certain weight    1  2    mock modular formsf  D   =   ∑   d  >  0    a   (  d  ,  D  )    q  d   ,    D  >  0    are given in terms of traces of cycle integrals of the   j  ‐function. Their result applies to those coefficients    a  (  d  ,  D  )    for which    d  D    is not a square. Recently, Bruinier, Funke, and Imamoḡlu employed a regularized theta lift to show that the coefficients    a  (  d  ,  D  )    for square    d  D    are traces of regularized integrals of the   j  ‐function. In the present paper, we provide an alternate approach to this problem. We introduce functions    j   m  ,  Q(for   Q   a quadratic form) which are related to the   j  ‐function and show, by modifying the method of Duke, Imamoḡlu, and Tóth, that the coefficients for which    d  D    is a square are traces of cycle integrals of the functions    j   m  ,  Q    .

Periods of the j ‐function along infinite geodesics and mock modular forms