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This paper presents a new profiling side-channel attack on the signature scheme CRYSTALS-Dilithium, which has been selected by the NIST as the new primary standard for quantum-safe digital signatures. This algorithm has a constant-time implementation with consideration for side-channel resilience. However, it does not protect against attacks that exploit intermediate data leakage. We exploit such a leakage on a vector generated during the signing process and whose costly protection by masking is a matter of debate. We design a template attack that enables us to efficiently predict whether a given coefficient in one coordinate of this vector is zero or not. Once this value has been completely reconstructed, one can recover, using linear algebra methods, part of the secret key that is sufficient to produce universal forgeries. While our paper deeply discusses the theoretical attack path, it also demonstrates the validity of the assumption regarding the required leakage model, from practical experiments with the reference implementation on an ARM Cortex-M4.