This is a question which has no "right" answer.

A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ that you put in the action of $GL_2^+(\mathbb{Q})$ determines which twist you get.

The normalisation that Diamond and Shurman use is the one that makes the eigenvalue of the double coset $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ on $\Pi_p$ correspond to the Fourier coefficient $a_p(f)$; while Bump's normalisation makes it correspond to $a_p(f) / p^{(k/2-1)}$.

From the perspective of the analytic theory of automorphic forms, Bump's choice is the "obviously right" one, since it makes $\Pi$ be unitary. Then you can find $\Pi$ as a subrepresentation of $L^2(GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A}))$ and the analytic theory works as it should. Since Bump's text emphasises the analytic theory of automorphic forms, this is the convention he chooses (and Paul Garrett's comment seems to be coming from the same viewpoint).

On the other hand, from the viewpoint of the algebraic theory (Galois representations, special values of L-functions, etc), the factor $k/2$ is extremely inconvenient, particularly when $k$ is odd. The $a_p$'s all lie in some common finite extension of $\mathbb{Q}$, but the eigenvalue of $\Pi_p$ has been multiplied by $p$ to a half-integer power; so the extension of $\mathbb{Q}$ generated by the eigenvalues is not finite, and correspondingly the critical values of the $L$-series are at half-integers rather than integers, meaning that the $L$-series of $\Pi$ cannot correspond to a motive. With Diamond and Shurman's normalisation, the finite part of $\Pi$ is definable over a number field, and its $L$-series is motivic. (This would work with any integer power of $\det$, but $\det^{k-1}$ is the minimal one which makes the Hecke eigenvalues algebraic integers.)

So the Diamond-Shurman normalisation is better for the algebraic theory, and the Bump normalisation for the analytic one.

There is a great quote from Deligne on this (possibly apocryphal): "Langlands is very convinced he knows what the square root of $p$ is. I have never been so sure."

This is a question which has no "right" answer.

A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ that you put in the action of $GL_2^+(\mathbb{Q})$ determines which twist you get.

The normalisation that Diamond and Shurman use is the one that makes the eigenvalue of the double coset $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ on $\Pi_p$ correspond to the Fourier coefficient $a_p(f)$; while Bump's normalisation makes it correspond to $a_p(f) / p^{(k/2-1)}$.

From the perspective of the analytic theory of automorphic forms, Bump's choice is the "obviously right" one, since it makes $\Pi$ be unitary. Then you can find $\Pi$ as a subrepresentation of $L^2(GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A}))$ and the analytic theory works as it should. Since Bump's text emphasises the analytic theory of automorphic forms, this is the convention he chooses (and Paul Garrett's comment seems to be coming from the same viewpoint).

On the other hand, from the viewpoint of the algebraic theory (Galois representations, special values of L-functions, etc), the factor $k/2$ is extremely inconvenient, particularly when $k$ is odd. The $a_p$'s all lie in some common finite extension of $\mathbb{Q}$, but the eigenvalue of $\Pi_p$ has been multiplied by $p$ to a half-integer power; so the extension of $\mathbb{Q}$ generated by the eigenvalues is not finite, and correspondingly the critical values of the $L$-series are at half-integers rather than integers, meaning that the $L$-series of $\Pi$ cannot correspond to a motive. With Diamond and Shurman's normalisation, the finite part of $\Pi$ is definable over a number field, and its $L$-series is motivic. (This would work with any integer power of $\det$, but $\det^{k-1}$ is the minimal one which makes the Hecke eigenvalues algebraic integers.)

So the Diamond-Shurman normalisation is better for the algebraic theory, and the Bump normalisation for the analytic one.

There is a great quote on this (attributed to Deligne, but possibly apocryphal): "Langlands is very convinced he knows what the square root of $p$ is. I have never been so sure."

This is a question which has no "right" answer.

A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ that you put in the action of $GL_2^+(\mathbb{Q})$ determines which twist you get.

The normalisation that Diamond and Shurman use is the one that makes the eigenvalue of the double coset $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ on $\Pi_p$ correspond to the Fourier coefficient $a_p(f)$, and has the functional equation of the $L$-series in the form $s \leftrightarrow k-s$; while Bump's normalisation makes it correspond to $a_p(f) / p^{(k/2-1)}$ and the functional equation is $s \leftrightarrow 1-s$.

From the perspective of the analytic theory of automorphic forms, Bump's choice is the "obviously right" one, since it makes $\Pi$ be unitary. Then you can find $\Pi$ as a subrepresentation of $L^2(GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A}))$ and the analytic theory works as it should. Since Bump's text emphasises the analytic theory of automorphic forms, this is the convention he chooses (and Paul Garrett's comment seems to be coming from the same viewpoint).

On the other hand, from the viewpoint of the algebraic theory (Galois representations, special values of L-functions, etc), the factor $k/2$ is extremely inconvenient, particularly when $k$ is odd. The $a_p$'s all lie in some common finite extension of $\mathbb{Q}$, but the eigenvalue of $\Pi_p$ has been multiplied by $p$ to a half-integer power; so the extension of $\mathbb{Q}$ generated by the eigenvalues is not finite, and correspondingly the critical values of the $L$-series are at half-integers rather than integers, meaning that the $L$-series of $\Pi$ cannot correspond to a motive. With Diamond and Shurman's normalisation, the finite part of $\Pi$ is definable over a number field, and its $L$-series is motivic. (This would work with any integer power of $\det$, but $\det^{k-1}$ is the minimal one which makes the Hecke eigenvalues algebraic integers.)

So the Diamond-Shurman normalisation is better for the algebraic theory, and the Bump normalisation for the analytic one.

There is a great quote from Deligne on this (possibly apocryphal): "Langlands is very convinced he knows what the square root of $p$ is. I have never been so sure."

This is a question which has no "right" answer.

A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ that you put in the action of $GL_2^+(\mathbb{Q})$ determines which twist you get.

The normalisation that Diamond and Shurman use is the one that makes the eigenvalue of the double coset $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ on $\Pi_p$ correspond to the Fourier coefficient $a_p(f)$; while Bump's normalisation makes it correspond to $a_p(f) / p^{(k/2-1)}$.

From the perspective of the analytic theory of automorphic forms, Bump's choice is the "obviously right" one, since it makes $\Pi$ be unitary. Then you can find $\Pi$ as a subrepresentation of $L^2(GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A}))$ and the analytic theory works as it should. Since Bump's text emphasises the analytic theory of automorphic forms, this is the convention he chooses (and Paul Garrett's comment seems to be coming from the same viewpoint).

On the other hand, from the viewpoint of the algebraic theory (Galois representations, special values of L-functions, etc), the factor $k/2$ is extremely inconvenient, particularly when $k$ is odd. The $a_p$'s all lie in some common finite extension of $\mathbb{Q}$, but the eigenvalue of $\Pi_p$ has been multiplied by $p$ to a half-integer power; so the extension of $\mathbb{Q}$ generated by the eigenvalues is not finite, and correspondingly the critical values of the $L$-series are at half-integers rather than integers, meaning that the $L$-series of $\Pi$ cannot correspond to a motive. With Diamond and Shurman's normalisation, the finite part of $\Pi$ is definable over a number field, and its $L$-series is motivic. (This would work with any integer power of $\det$, but $\det^{k-1}$ is the minimal one which makes the Hecke eigenvalues algebraic integers.)

So the Diamond-Shurman normalisation is better for the algebraic theory, and the Bump normalisation for the analytic one.

There is a great quote from Deligne on this (possibly apocryphal): "Langlands is very convinced he knows what the square root of $p$ is. I have never been so sure."