equation solving - I can't understand why "FindRoot::nlnum:" shows up in my code

I want to solve a system of equations using Findroot as follows:

Clear[a]; Clear[c]; n = 2;
SysEqn1 = Table[a[i] + I Sum[(PolyLog[1, E^(I (c[j] - a[i] + 0.5))] + 

    PolyLog[1, E^(I (c[j] - a[i] + (2 \[Pi] - 1.2)))] - 
    PolyLog[1, E^(I (c[j] - a[i] - 0.3))] - 
    PolyLog[1, E^(I (c[j] - a[i] - 0.4))]), {i, n}], {j, n}];
SysEqn2 = Table[c[i] + I Sum[(PolyLog[1, E^(I (c[j] - a[i] + 0.5))] + 
    PolyLog[1, E^(I (c[j] - a[i] + (2 \[Pi] - 1.2)))] - 
    PolyLog[1, E^(I (c[j] - a[i] - 0.3))] - 
    PolyLog[1, E^(I (c[j] - a[i] - 0.4))]), {i, n}], {j, n}];
SysEqn = Join[SysEqn1, SysEqn2]; 
startingValues1 = Table[{a[i], -1 + 2 i/n}, {i, n}]; 
startingValues2 = Table[{c[i], -0.9 + 2 i/n}, {i, n}]; 

starting = Join[startingValues1, startingValues2];
FindRoot[SysEqn, starting]

The error message from the above code is as fo

FindRoot::nlnum: "The function value {a[i]+(0. +1.\ I)\ (1.\ Log[1. +Times[<<2>>]]+1.\ Log[1. +Times[<<2>>]]-1.\ Log[1. +Times[<<2>>]]-1.\ Log[1. +Times[<<2>>]]+1.\ Log[1. +Times[<<2>>]]+1.\ Log[1. +Times[<<2>>]]-1.\ Log[1. +Times[<<2>>]]-1.\ Log[1. +Times[<<2>>]]),a[i]+(0. +1.\ I)\ (<<1>>),<<1>>,c[i]+(0. +1.\ I)\ (1.\ Log[1. +Times[<<2>>]]+<<11>>)} is not a list of numbers with dimensions {4} at {a[1],a[2],c[1],c[2]} = {0.,1.,0.1,1.1}."

For n=2 case, I have four equations so I think the number of starting variables that should be determined is also four. But for some reason my code doesn't work... I would really appreciate if you can help me out with this problem!

Answer

First of all, there seems to be a bug:

PolyLog[1, E^(I (c[1] - a[2] + (2 π - 1.2)))]

(*  -1. Log[1 - 2.71828^((0. + 1. I) (5.08319 - 1. a[2.] + c[1.]))]  *)

Note the indices have been converted from Integer to Real numbers. This can be easily fixed with NHoldAll.

Second, there's an a[i] and c[i] at the beginning of Table in each system, but the Table iterator is j. Looking at the image posted in the original form of the question, it seems the j iterator for the Sum was misplaced in the edit.

ClearAll[a];
ClearAll[c];
SetAttributes[a, NHoldAll];
SetAttributes[c, NHoldAll];
n = 2;
SysEqn1 = Table[a[i] + I Sum[(PolyLog[1, E^(I (c[j] - a[i] + 0.5))] + 

        PolyLog[1, E^(I (c[j] - a[i] + (2 π - 1.2)))] - 
        PolyLog[1, E^(I (c[j] - a[i] - 0.3))] - 
        PolyLog[1, E^(I (c[j] - a[i] - 0.4))]), {j, n}], {i, n}];
SysEqn2 = Table[c[i] + I Sum[(PolyLog[1, E^(I (c[j] - a[i] + 0.5))] + 
        PolyLog[1, E^(I (c[j] - a[i] + (2 π - 1.2)))] - 
        PolyLog[1, E^(I (c[j] - a[i] - 0.3))] - 
        PolyLog[1, E^(I (c[j] - a[i] - 0.4))]), {j, n}], {i, n}];
SysEqn = Join[SysEqn1, SysEqn2];
startingValues1 = Table[{a[i], -1 + 2 i/n}, {i, n}];
startingValues2 = Table[{c[i], -0.9 + 2 i/n}, {i, n}];

starting = Join[startingValues1, startingValues2];

FindRoot[SysEqn, starting]
(*
  {a[1] -> 1.8484 + 0.907529 I, a[2] -> 3.0128 + 1.14394 I, 
   c[1] -> 1.8484 + 0.907529 I, c[2] -> 3.0128 + 1.14394 I}
*)