Despite advancements, fine-tuning Large Language Models (LLMs) remains costly due to the extensive parameter count and substantial data requirements for model generalization. Accessibility to computing resources remains a barrier for the open-source community. To address this challenge, we propose the In2Core algorithm, which selects a coreset by analyzing the correlation between training and evaluation samples with a trained model. Notably, we assess the model's internal gradients to estimate this relationship, aiming to rank the contribution of each training point. To enhance efficiency, we propose an optimization to compute influence functions with a reduced number of layers while achieving similar accuracy. By applying our algorithm to instruction finetuning data of LLMs, we can achieve similar performance with just 50% of the training data. Meantime, using influence functions to analyze model coverage to certain testing samples could provide a reliable and interpretable signal on the training set's coverage of those test points.