There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomial only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as a sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series.

My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic quadratic polynomial with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledge of theory of Modular Forms is not so good, so I would ask you advice. Thank you in advance.

There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomials only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as a sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series.

My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic quadratic polynomial with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledge of theory of Modular Forms is not so good, so I would ask you an advice. Thank you in advance.

There is a huge amount of research dealing with analysis of representation of integers by quadratic forms only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series.

My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic forms with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledhe of theory of Modular Forms is not so good, so I would ask you advice. Thank you in advance.

There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomial only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as a sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series.

My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic quadratic polynomial with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledge of theory of Modular Forms is not so good, so I would ask you advice. Thank you in advance.