In a manner that shows ambition, hope, or striving toward a goal; with aspiring qualities or intent.
From 'aspiring' (present participle of 'aspire') + '-ly' (adverb suffix). The '-ly' suffix transforms adjectives into adverbs describing how or in what manner something is done.
When someone looks 'aspiringly' at something, they're showing that hopeful, ambitious expression—it's a great word for creative writing because it captures that particular kind of determined dreaming.
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