
> 
> Let $A\_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such that $1<tuv|N$. For any such triple, define
>  \begin{equation}
> E\_2^{\psi,\varphi,t}(\tau)=\begin{cases}
> E\_2^{\psi,\varphi}(t\tau)& \text{unless $\psi=\varphi=\textbf{1}\_1$,} \\
> E\_2^{\textbf{1}\_1,\textbf{1}\_1}(\tau)-tE\_2^{\textbf{1}\_1,\textbf{1}\_1}(t\tau)& \text{if $\psi=\varphi=\textbf{1}\_1$.} 
> \end{cases} \nonumber
> \end{equation}
>  where $E\_2^{\psi,\varphi}$ is the Eisenstein series of weight $2$ attached to $(\psi,\varphi)$ (see the book of Diamond and Shurman : Introduction to Modular Forms p.133).
> 
> 
> 


We learn in this book that the set $\mathcal{B}:=\left\{E\_2^{\psi,\varphi,t}:(\psi,\varphi,t)\in A\_{N,2}, \psi\varphi=\textbf{1}\_N \right\}$ represents a basis of the Eisenstein space of weight $2$ with respect to $\Gamma\_0(N)$. Here is my issue :



> 
> I don't clearly understand the equality $\psi \varphi=\textbf{1}\_N$ in the definition of $\mathcal{B}$ : for me, if $\psi$ and $\varphi$ are two Dirichlet characters modulo $u$ and $v$, then $\psi\varphi$ is a Dirichlet character modulo $\operatorname{lcm}(u,v)$, and in particular $\textbf{1}\_1 \textbf{1}\_1=\textbf{1}\_1\neq \textbf{1}\_N$ if $N\neq 1$. Following this, since $N>1$, $E\_2^{\textbf{1}\_1,\textbf{1}\_1,t}$ never appears as an element of the above basis $\mathcal{B}$.
> 
> 
> 


In my view, the condition $\psi \varphi=\textbf{1}\_N$ should be replaced by the relation $\psi \varphi \textbf{1}\_N=\textbf{1}\_N$.


So where am I wrong? Many thanks for your help!


Ps: I asked this question on Math StackExchange but there is no answer yet.



Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?
The first part of Theorem 4.6.2 in the book says that $\left\{E\_2^{\psi,\varphi,t}:(\psi,\varphi,t)\in A\_{N,2}\right\}$ represents a basis of the Eisenstein space of weight $2$ with respect to $\Gamma\_1(N)$. As $E\_2^{\psi,\varphi,t}$ as a modular form for $\Gamma\_0(N)$ has nebentypus $\psi\phi$ regarded as a modulo $N$ Dirichlet character, in the second part of the theorem the condition $\psi\phi=\chi$ is meant to be an equation of modulo $N$ Dirichlet characters, i.e. it really means $\psi\phi \textbf{1}\_N=\chi$ as an equation of functions $\mathbf{Z}\to\mathbf{C}$.


To summarize, what you think is right, and this is what the authors of the book had in mind, too.


