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Abstract

Let $f$ be a new cusp form on $\Gamma_0(N)$ of even weight $k+2\geq 2$. Suppose that there is a prime $p\parallel N$ and that we may write $N=pN^{+}N^{-}$, where $N^{-}$ is the squarefree product of an even number of primes. There is a Darmon style ${\cal L}$-invariant ${\cal L}^{N^{-}}(f)$ attached to this factorization, which is the Orton ${\cal L}$-invariant when $N^{-}=1$. We prove that ${\cal L}^{N^{-}}(f)$ does not depend on the chosen factorization of $N$ and it is equal to the other known ${\cal L}$-invariants. We also give a formula for the computation of the logarithmic $p$-adic Abel-Jacobi image of the Darmon cycles. This formula is crucial for the computations of the derivatives of the $p$-adic $L$-functions of the weight variable attached to a real quadratic field $K/\Bbb{Q}$ such that the primes dividing $N^{+}$ are split and the primes dividing $pN^{-}$ are inert.

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