How is the Petersson norm of a quaternionic modular form defined?


*Background:* In [Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms](https://arxiv.org/pdf/1911.05019.pdf), section 3.3, it is written "We normalize $f\_B$ requiring its Petersson norm to be $1$ (the Petersson product being just a finite sum in this case).", but no reference to the definition is given.


My guess would be to sum over the squares finitely many values of the QMF weighted by the number of units of the left order of the right ideal classes divided by $2$: $$\|f\_B\|^2 = \sum\_{[I]}f\_B([I])^2 \cdot (\#\mathcal{O}\_l([I])^\times/2)$$


But with this definition, the element $a(f)$ in *loc. cit.* seems not to be rational for a classical modular form $f$ with $L(f,1) \neq 0$. (Concretely, for the newform of level $11$.)



Petersson norms of quaternionic modular forms
"Of Petersson norm one" in the paper you mention means "of norm one" in the sense of Theorem 7.1 of S. Zhang's paper "Gross-Zagier formula for GL(2) II".
With this normalisation of the Jacquet-Langlands transfer $f\_B$, as you write, $a(f)$ is not necessarily rational.


On the other hand, outside the statement of Theorem 3.4 of [Tamiozzo, https://arxiv.org/pdf/1911.05019.pdf], a different normalisation of $f\_B$, as in Remark 3.5, is used to define $a(f)$ (see also Sections 6.1, 6.2 of W. Zhang's "Selmer groups and the indivisibility of Heegner points" for a related discussion).


