Large language models are transforming research on machine learning while galvanizing public debates. Understanding not only when these models work well and succeed but also why they fail and misbehave is of great societal relevance. We propose to turn the lens of computational psychiatry, a framework used to computationally describe and modify aberrant behavior, to the outputs produced by these models. We focus on the Generative Pre-Trained Transformer 3.5 and subject it to tasks commonly studied in psychiatry. Our results show that GPT-3.5 responds robustly to a common anxiety questionnaire, producing higher anxiety scores than human subjects. Moreover, GPT-3.5's responses can be predictably changed by using emotion-inducing prompts. Emotion-induction not only influences GPT-3.5's behavior in a cognitive task measuring exploratory decision-making but also influences its behavior in a previously-established task measuring biases such as racism and ableism. Crucially, GPT-3.5 shows a strong increase in biases when prompted with anxiety-inducing text. Thus, it is likely that how prompts are communicated to large language models has a strong influence on their behavior in applied settings. These results progress our understanding of prompt engineering and demonstrate the usefulness of methods taken from computational psychiatry for studying the capable algorithms to which we increasingly delegate authority and autonomy.