HYDERABAD, India — At least 100,000 people gathered Saturday in Hyderabad, India’s technology hub, to protest Prime Minister Narendra Modi and a new law they say will strip the country of its secular foundations, maintaining steady pressure on the government as demonstrations entered their fourth week.

The protests have drawn massive crowds across India, with more than 200,000 people gathering in Kochi city, in the southern state of Kerala, on New Year’s Day. And in Delhi, hundreds continued to camp out on a vital stretch of highway that links the capital to its suburbs, bristling against one of the city’s coldest winters in decades.

While the protests are the biggest threat yet to Mr. Modi’s tenure in office, they may also be the beginning of a deeper political and social shift in India. From the start the protests have attracted Indians across political stripes and creeds. But with India’s Muslims spearheading the demonstrations this past week, the 200-million strong minority showed it can organize as a formidable force to check Mr. Modi’s Hindu nationalist government.

The protests began in December when the government passed a law that uses religion as a criterion for determining whether illegal migrants in India can be fast-tracked for citizenship. The measure favors members of all South Asia’s major religions except Islam, India’s second largest faith. Muslims worry that the law will be coupled with a citizenship test and used to strip them of their Indian nationality.