In their 1984 paper on “Modular forms with rational periods” Kohnen and Zagier found that products of at most two level 1 Eisenstein Series \( E_l \) and \( E_{k-l} \) span all spaces of (level 1) modular forms. Their result is based on a formula for Rankin convolutions of newforms and $ E_l E_{k-l} $. It therefore provides a deep connection to periods of modular forms. We present an analogue of Kohnen's and Zagier's result in the level aspect. As opposed to previous studies, we fix \( l \), and let very the level of Eisenstein series. The crucial idea to our result is to employ vector valued modular forms. Discussion of the result includes “modular forms as functors” and a way to recover several classical constructions in the vector valued setting.

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