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LetD  &lt;  0    ${D &amp;lt; 0}$    be an odd fundamental discriminant and  q  be a prime number which splits inℚ  (   D   )    ${{\mathbb {Q}}(\sqrt{D})}$   . Given a suitable smooth function  f  supported on[  X  ,  2  X  ]    ${[X, 2X]}$    for  X  ≥ 1, we establish a uniform bound inX  ,  D    ${X, D}$    and  q  for∑   c  ≡  0    (  mod    q  )W  h    (  D  ;  c  )   f   (  c  )    ,    $ {\sum _{c \equiv 0 \hspace*{1.42262pt}(\operatorname{mod} ~q)}W_h(D;c)f(c)}, $    whereW  h    (  D  ;  c  )   :  =   ∑   b(  mod    2  c  )   ,b  2   ≡  D(  mod    4  c  )eh  b    2  c,    h  ∈  ℤ  ,    e   (  z  )   :  =   e   2  π  i  z    ,    $ {W_h(D;c) := \sum _{b \hspace*{1.42262pt}(\operatorname{mod} ~2c),\, b^2 \equiv D \hspace*{1.42262pt}(\operatorname{mod} ~4c)}e\biggl (\frac{hb}{2c} \biggr )}, \quad h \in {\mathbb {Z}}, \quad e(z):= e^{2 \pi i z}, $    is the Weyl sum for roots of the quadratic congruencex  2   ≡  D(  mod4  c   )${x^2 \equiv D~(\text{mod}~{4c})}$   . We use this result to study several problems of arithmetic interest, including level-aspect versions of equidistribution of quadratic roots, and the asymptotic distribution of traces of CM values of weakly holomorphic modular functions of level  q . By work of Zagier and Bruinier–Funke, the generating functions for these traces are weight 3/2 weakly holomorphic modular forms of level4  q    ${4q}$    satisfying Kohnen's plus space condition.

Hybrid bounds for quadratic Weyl sums and arithmetic applications