Some Maass form can be written ($K\_{iR}$ is the K-Bessel function):


$$f(x+iy)=\sum\_{n \ne 0}^{\infty} a\_n \sqrt{y} \;K\_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$


with the $a\_n$ multiplicative, but inversly if I fix multiplicative coefficients, for example $a\_n=\frac{\chi(n)}{\sqrt{|n|}}$ can I expect the function defined by following sum to have some properties of Maass forms?


$$g(x+iy)=\sum\_{n\ne 0}^{\infty} \frac{\chi(n)}{\sqrt{|n|}} \sqrt{y} \;K\_{iR}(2\pi |n|y) \; e^{2 i\pi nx}$$


For example to have $g(\frac{az+b}{cz+d})$ linked with $g(z)$ for certain $a,b,c,d$ like for a Maass form ? Or there is nothing to expect ?


Any reference on the demonstration of multiplicative property of Maass form coefficients ? (I found plenty references for classic modular forms but less for Maass forms and generaly there is no detail)



Maass form properties and their fourier coefficients
A few things.


* All Maass forms can be written in the Bessel form that you mention. It is deduced from the growth condition of the Fourier coefficients of an arbitrary Maass forms.
* If you want to twist a Maass form $f$ by a primitive Dirichlet character $\chi$, the natural object to consider is the twisted L-function associated to the Maass form:


$$L(s,f,\chi)=\sum \frac{\chi(n)a\_n}{n^s}$$


* If you directly modify the coefficients of the Maass form, I think you shouldn't except it to be automorphic anymore (that is, automorphic with probability 0). See znt's comment above. After all, all the data of the Maass form is carried in its coefficients, and you have deleted it (the $a\_n$) and you have replaced it with unreleated data from a Dirichlet coefficient mod n.
* I can't think of any reference. One usually sees the proof for modular forms, and then for all automorphic forms. But the idea is still the same (see again znt's comment).


