Generating poetry has become a popular application of LLMs, perhaps especially of OpenAI's widely-used chatbot ChatGPT. What kind of poet is ChatGPT? Does ChatGPT have its own poetic style? Can it successfully produce poems in different styles? To answer these questions, we prompt the GPT-3.5 and GPT-4 models to generate English-language poems in 24 different poetic forms and styles, about 40 different subjects, and in response to 3 different writing prompt templates. We then analyze the resulting 5.7k poems, comparing them to a sample of 3.7k poems from the Poetry Foundation and the Academy of American Poets. We find that the GPT models, especially GPT-4, can successfully produce poems in a range of both common and uncommon English-language forms in superficial yet noteworthy ways, such as by producing poems of appropriate lengths for sonnets (14 lines), villanelles (19 lines), and sestinas (39 lines). But the GPT models also exhibit their own distinct stylistic tendencies, both within and outside of these specific forms. Our results show that GPT poetry is much more constrained and uniform than human poetry, showing a strong penchant for rhyme, quatrains (4-line stanzas), iambic meter, first-person plural perspectives (we, us, our), and specific vocabulary like "heart," "embrace," "echo," and "whisper."