{'fraction_non_alphanumeric': 0.018475750577367205, 'fraction_numerical': 0.0, 'mean_word_length': 5.78125, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hecke’s modular forms by functional equations

We investigate the interplay between multiplicative Hecke operators, including bad primes, and the characterization of modular forms studied by Hecke. The operators are applied on periodic functions, which lead to functional equations characterizing certain eta-quotients. This can be considered as a prototype for functional equations in the more general context of Borcherds products.