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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.

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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.

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