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The displayed definition uses iterated Eisenstein summation in weight two: the sum in $n$ is evaluated before the sum in $m$, with the single $(0,0)$ term omitted. This order is essential. The two-dimensional lattice series does not have absolute convergence, so arbitrary rearrangement would not be justified.

For noninteger $w$, the cosecant partial-fraction identity isFor completeness, apply the residue theorem to $\pi \cot (\pi \zeta )/(\zeta -w{)}^2$ on squares with large half-integer sides. The cotangent is bounded on the contours and the integral is $O(R^{-1})$. Its residues at the integers are $(n-w{)}^{-2}$ and its residue at $w$ is $-{\pi }^2{\csc }^2(\pi w)$, proving the formula. If $\mathrm{Im}w>0$, the geometric-series expression $\cot (\pi w)=-i(1+2{\sum }_{r\ge 1}{e}^{2\pi irw})$, differentiated termwise, gives the cotangent partial-fraction Fourier kernel

Put $q=e^{2\pi iz}$ with $z$ in the complex upper half-plane. For positive $m$, this gives $-4{\pi }^2{\sum }_{r\ge 1}r{q}^{mr}$. Negative $m$ gives the same value, by replacing $n$ with $-n$ in its inner sum. The $m=0$ row is ${\sum }_{n\ne 0}{n}^{-2}={\pi }^2/3$, by the Basel problem. The resulting series in $m,r$ does have absolute convergence, locally uniformly in $z$, so collecting the coefficient at $q^{\ell }$ is legitimate:The coefficient is the sum-of-divisors function, since $r$ runs over the positive divisors of $\ell$. ThusThere is no conflict with vanishing of weight-two level-one modular forms: the Eisenstein series of weight two has an anomalous transformation term, so it is not a weight-two modular form.

The original PDF has the summation condition $\gcd (n,N)=1$; the TeX's $\gcd (n,n)=1$ is a transcription error. There is also an actual missing hypothesis in the PDF's coefficient formula: that simplified formula requires a primitive Dirichlet character. We first derive a formula valid for every character, and then show both the primitive specialization and a counterexample to the unrestricted version.

For $k>2$, the character-twisted Eisenstein series converges absolutely and locally uniformly on the complex upper half-plane. On a compact subset, $|mz+n|$ is bounded below by a positive constant times $(m^2+n^2{)}^{1/2}$, and the corresponding two-dimensional lattice sum converges. Changing $n$ to $n+mN$ provesThe condition $\chi (-1)=(-1{)}^k$ makes the terms for $(m,n)$ and $(-m,-n)$ equal. The terms with $m=0$ contribute $2L(\chi ,k)$, and all other terms are twice the sum over $m\ge 1$.

For $\mathrm{Im}w>0$, the cotangent identity andgive, after $k-1$ differentiations,Differentiation is justified by locally uniform convergence. This is the cotangent partial-fraction Fourier kernel.

Write $n=a+Nt$, with $a$ running through the unit classes modulo $N$, and define the finite Fourier transformHere $g(\chi )$ is the Gauss sum of a Dirichlet character.
Applying the kernel with $w=(mz+a)/N$ yieldsThe double series converges absolutely: $|S_{\chi }(r)|\le \phi (N)$ and the exponential decay controls ${\sum }_{m,r}{r}^{k-1}{e}^{-2\pi mr\mathrm{Im}z/N}$. Grouping the terms with $mr=n$ proves the general Fourier expansion of a character-twisted Eisenstein series:

If $r$ is a unit modulo $N$, substitution $a\mapsto r^{-1}a$ gives $S_{\chi }(r)=\chi (r{)}^{-1}g(\chi )$. Suppose now that $\chi$ is primitive and $r$ is a nonunit. Choose a prime $p\mid \gcd (r,N)$. Reduction of units modulo $N$ onto units modulo $N/p$ is surjective. Primitivity means that $\chi$ is nontrivial on its kernel, so there is a unit $u\equiv 1(\mathrm{mod}N/p)$ with $\chi (u)\ne 1$. Since $ru\equiv r(\mathrm{mod}N)$, substitution by $u$ forces $S_{\chi }(r)=\chi (u{)}^{-1}{S}_{\chi }(r)$ and thus $S_{\chi }(r)=0$. This proves the finite Fourier transform of a primitive Dirichlet character identity, and consequentlyThe inverse-character notation in the question is understood to mean $\overline{\chi }(r)$, extended by zero on nonunits; literal inversion of $\chi (r)=0$ would be undefined.

For a counterexample without primitivity, take $N=2$, the principal character, and $k=4$. Then $S_{\chi }(r)=(-1{)}^r$, so $g(\chi )=-1$ but $S_{\chi }(2)=1$. At $n=2$, the general formula gives $c_2=7A_4/8$, whereas the printed simplified formula, interpreted as zero on nonunits, gives $-A_4/8$. Hence the printed coefficient formula is false for arbitrary imprimitive characters; the general boxed formula above supplies the correction.