Finite Group Databasebeta

← Group #13161833

Character table — Group #13161833

Order 1728.

C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C23C24C25C26C27C28C29C30C31C32C33C34C35C36C37C38C39C40C41C42C43C44C45C46C47C48C49C50C51C52C53C54C55C56C57C58C59C60C61C62C63C64C65C66C67C68C69C70C71C72C73C74C75C76C77C78C79C80C81C82C83C84C85C86C87C88C89C90C91C92C93C94C95C96C97C98C99C100C101C102C103C104C105C106C107C108C109C110C111C112C113C114C115C116C117C118C119C120C121C122C123C124C125C126C127C128C129C130C131C132C133C134C135C136C137C138C139C140C141C142C143C144C145C146C147C148C149C150C151C152C153C154C155C156C157C158C159C160C161C162C163C164C165C166C167C168C169C170C171C172C173C174C175C176C177C178C179C180C181C182C183C184C185C186C187C188C189C190C191C192
size1111111122222222222222222222222222222222222244444444444444444444444444444444444444444444444444444444444444444444444444444444444444449999999918181818181818181818181818181818181818181818181818181818181818181818181836363636363636363636363636363636
order1222222233344446666666666666666666661212121212121212333336666666666666666666666666666666666612121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121244444444444444444444444444441212121212121212121212121212121212121212121212121212121212121212
χ1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
χ2111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
χ3111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-1
χ4111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111
χ511111111111-1-1-1-1111111111111111111111-1-1-1-1-1-1-1-11111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-11111-1-1-1-11111-1-1-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-11111-1-1-1-1
χ611111111111-1-1-1-1111111111111111111111-1-1-1-1-1-1-1-11111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-11111
χ711111111111-1-1-1-1111111111111111111111-1-1-1-1-1-1-1-11111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-1-11111
χ811111111111-1-1-1-1111111111111111111111-1-1-1-1-1-1-1-11111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-111111111-1-1-1-111111111-1-1-1-1
χ91111-1-1-1-111111-1-11111-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111-1-1-1-111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-111-1-1ii−i−iii−i−iii−i−iii−i−i1111-1-1-1-11111-1-1-1-1ii−i−iii−i−iii−i−iii−i−i
χ101111-1-1-1-111111-1-11111-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111-1-1-1-111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-111-1-1−i−iii−i−iii−i−iii−i−iii1111-1-1-1-11111-1-1-1-1−i−iii−i−iii−i−iii−i−iii
χ111111-1-1-1-111111-1-11111-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111-1-1-1-111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-111ii−i−iii−i−i−i−iii−i−iii-1-1-1-11111-1-1-1-11111ii−i−iii−i−i−i−iii−i−iii
χ121111-1-1-1-111111-1-11111-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111-1-1-1-111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-111−i−iii−i−iiiii−i−iii−i−i-1-1-1-11111-1-1-1-11111−i−iii−i−iiiii−i−iii−i−i
χ131111-1-1-1-1111-1-1111111-1-1-1-1-1-1-1-111111-1-1-1-1-1-1-1-11111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111111111111111111-1-1-1-1-1-111ii−i−i−i−iiiii−i−i−i−iii1111-1-1-1-1-1-1-1-11111ii−i−i−i−iiiii−i−i−i−iii
χ141111-1-1-1-1111-1-1111111-1-1-1-1-1-1-1-111111-1-1-1-1-1-1-1-11111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111111111111111111-1-1-1-1-1-111−i−iiiii−i−i−i−iiiii−i−i1111-1-1-1-1-1-1-1-11111−i−iiiii−i−i−i−iiiii−i−i
χ151111-1-1-1-1111-1-1111111-1-1-1-1-1-1-1-111111-1-1-1-1-1-1-1-11111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111111111111111-1-1-1-1111111-1-1ii−i−i−i−iii−i−iiiii−i−i-1-1-1-111111111-1-1-1-1ii−i−i−i−iii−i−iiiii−i−i
χ161111-1-1-1-1111-1-1111111-1-1-1-1-1-1-1-111111-1-1-1-1-1-1-1-11111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111-1-1-1-11111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111111111111111-1-1-1-1111111-1-1−i−iiiii−i−iii−i−i−i−iii-1-1-1-111111111-1-1-1-1−i−iiiii−i−iii−i−i−i−iii
χ171-11-111-1-1111i−ii−i11-1-11111-1-1-1-1-1-1-11-111-1-1ii−i−iii−i−i1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1ii−i−iii−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−i11-1-111-1-1i−ii−i1-11-1i−ii−i1-11-1i−ii−i11-1-111-1-1ii−i−iii−i−i1-11-1i−ii−i1-11-1i−ii−i
χ181-11-111-1-1111i−ii−i11-1-11111-1-1-1-1-1-1-11-111-1-1ii−i−iii−i−i1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1ii−i−iii−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−i11-1-111-1-1i−ii−i-11-11−ii−ii-11-11−ii−ii11-1-111-1-1ii−i−iii−i−i-11-11−ii−ii-11-11−ii−ii
χ191-11-111-1-1111i−ii−i11-1-11111-1-1-1-1-1-1-11-111-1-1ii−i−iii−i−i1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1ii−i−iii−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−i-1-111-1-111−ii−ii1-11-1i−ii−i-11-11−ii−ii-1-111-1-111−i−iii−i−iii1-11-1i−ii−i-11-11−ii−ii
χ201-11-111-1-1111i−ii−i11-1-11111-1-1-1-1-1-1-11-111-1-1ii−i−iii−i−i1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1ii−i−iii−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−i-1-111-1-111−ii−ii-11-11−ii−ii1-11-1i−ii−i-1-111-1-111−i−iii−i−iii-11-11−ii−ii1-11-1i−ii−i
χ211-11-111-1-1111−ii−ii11-1-11111-1-1-1-1-1-1-11-111-1-1−i−iii−i−iii1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1−i−iii−i−iii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii11-1-111-1-1−ii−ii1-11-1−ii−ii1-11-1−ii−ii11-1-111-1-1−i−iii−i−iii1-11-1−ii−ii1-11-1−ii−ii
χ221-11-111-1-1111−ii−ii11-1-11111-1-1-1-1-1-1-11-111-1-1−i−iii−i−iii1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1−i−iii−i−iii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii11-1-111-1-1−ii−ii-11-11i−ii−i-11-11i−ii−i11-1-111-1-1−i−iii−i−iii-11-11i−ii−i-11-11i−ii−i
χ231-11-111-1-1111−ii−ii11-1-11111-1-1-1-1-1-1-11-111-1-1−i−iii−i−iii1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1−i−iii−i−iii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii-1-111-1-111i−ii−i1-11-1−ii−ii-11-11i−ii−i-1-111-1-111ii−i−iii−i−i1-11-1−ii−ii-11-11i−ii−i
χ241-11-111-1-1111−ii−ii11-1-11111-1-1-1-1-1-1-11-111-1-1−i−iii−i−iii1111111111-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1−i−iii−i−iii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii-1-111-1-111i−ii−i-11-11i−ii−i1-11-1−ii−ii-1-111-1-111ii−i−iii−i−i-11-11i−ii−i1-11-1−ii−ii
χ251-11-1-1-111111i−i−ii11-1-1-1-1-1-11111-1-1-11-1-1-111ii−i−i−i−iii1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1ii−i−i−i−iiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiii11-1-1-1-111i−i−iii−i−ii-111-1i−i−ii-111-111-1-1-1-111ii−i−i−i−iiii−i−ii-111-1i−i−ii-111-1
χ261-11-1-1-111111i−i−ii11-1-1-1-1-1-11111-1-1-11-1-1-111ii−i−i−i−iii1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1ii−i−i−i−iiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiii11-1-1-1-111i−i−ii−iii−i1-1-11−iii−i1-1-1111-1-1-1-111ii−i−i−i−iii−iii−i1-1-11−iii−i1-1-11
χ271-11-1-1-111111i−i−ii11-1-1-1-1-1-11111-1-1-11-1-1-111ii−i−i−i−iii1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1ii−i−i−i−iiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiii-1-11111-1-1−iii−ii−i−ii-111-1−iii−i1-1-11-1-11111-1-1−i−iiiii−i−ii−i−ii-111-1−iii−i1-1-11
χ281-11-1-1-111111i−i−ii11-1-1-1-1-1-11111-1-1-11-1-1-111ii−i−i−i−iii1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1ii−i−i−i−iiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiii-1-11111-1-1−iii−i−iii−i1-1-11i−i−ii-111-1-1-11111-1-1−i−iiiii−i−i−iii−i1-1-11i−i−ii-111-1
χ291-11-1-1-111111−iii−i11-1-1-1-1-1-11111-1-1-11-1-1-111−i−iiiii−i−i1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1−i−iiiii−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i11-1-1-1-111−iii−ii−i−ii1-1-11i−i−ii1-1-1111-1-1-1-111−i−iiiii−i−ii−i−ii1-1-11i−i−ii1-1-11
χ301-11-1-1-111111−iii−i11-1-1-1-1-1-11111-1-1-11-1-1-111−i−iiiii−i−i1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1−i−iiiii−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i11-1-1-1-111−iii−i−iii−i-111-1−iii−i-111-111-1-1-1-111−i−iiiii−i−i−iii−i-111-1−iii−i-111-1
χ311-11-1-1-111111−iii−i11-1-1-1-1-1-11111-1-1-11-1-1-111−i−iiiii−i−i1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1−i−iiiii−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i-1-11111-1-1i−i−iii−i−ii1-1-11−iii−i-111-1-1-11111-1-1ii−i−i−i−iiii−i−ii1-1-11−iii−i-111-1
χ321-11-1-1-111111−iii−i11-1-1-1-1-1-11111-1-1-11-1-1-111−i−iiiii−i−i1111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1−i−iiiii−i−i−i−i−i−i−i−i−i−i−i−iiiiiiiiiiiiiiiiiiiii−i−i−i−i−i−i−i−i−i−i-1-11111-1-1i−i−ii−iii−i-111-1i−i−ii1-1-11-1-11111-1-1ii−i−i−i−iii−iii−i-111-1i−i−ii1-1-11
χ332222222222-12222222222-1-122-1-122-1-1-12222-1-1-1-1-1-1-1-12-1-1-1-12-1-1-1-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-12-1-1-1-12222222222-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-12222222222220000000000000000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-10000000000000000
χ342222222222-12222222222-1-122-1-122-1-1-12222-1-1-1-1-1-1-1-12-1-1-1-12-1-1-1-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-12-1-1-1-12222222222-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2000000000000000011111111111111110000000000000000
χ352222222222-1-2-2-2-2222222-1-122-1-122-1-1-12222111111112-1-1-1-12-1-1-1-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-12-1-1-1-1-2-2-2-2-2-2-2-2-2-211111111-2-211111111-2-211111111-2-21111111122222222-2-2-2-20000000000000000-1-1-1-1-1-1-1-1111111110000000000000000
χ362222222222-1-2-2-2-2222222-1-122-1-122-1-1-12222111111112-1-1-1-12-1-1-1-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-12-1-1-1-1-2-2-2-2-2-2-2-2-2-211111111-2-211111111-2-211111111-2-211111111-2-2-2-2-2-2-2-22222000000000000000011111111-1-1-1-1-1-1-1-10000000000000000
χ37222222222-1222222-12-12-1222-1222-12222-12-122222222-12-1-1-1-12-1-1-1-12-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-12-1-1-12-12-12-12-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-10000000000000000000022222222000000000000000000000000-1-1-1-1-1-1-1-1
χ38222222222-1222222-12-12-1222-1222-12222-12-122222222-12-1-1-1-12-1-1-1-12-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-12-1-1-12-12-12-12-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-100000000000000000000-2-2-2-2-2-2-2-200000000000000000000000011111111
χ39222222222-12-2-2-2-22-12-12-1222-1222-12222-12-1-2-2-2-2-2-2-2-2-12-1-1-1-12-1-1-1-12-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-12-1-1-1-21-21-21-2111-2-211111111-2-211111111-2-211111111-2-2111111000000000000000000002222-2-2-2-2000000000000000000000000-1-1-1-11111
χ40222222222-12-2-2-2-22-12-12-1222-1222-12222-12-1-2-2-2-2-2-2-2-2-12-1-1-1-12-1-1-1-12-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-12-1-1-1-21-21-21-2111-2-211111111-2-211111111-2-211111111-2-211111100000000000000000000-2-2-2-222220000000000000000000000001111-1-1-1-1
χ4122222222-1222222-12-12-1222-1222-12222-12-1222222222-1-12-1-1-1-12-1-1-1-12-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-12-1-1-12-12-12-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-100000000000022222222000000000000000000000000-1-1-1-1-1-1-1-100000000
χ4222222222-1222222-12-12-1222-1222-12222-12-1222222222-1-12-1-1-1-12-1-1-1-12-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-12-1-1-12-12-12-12-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1000000000000-2-2-2-2-2-2-2-20000000000000000000000001111111100000000
χ4322222222-122-2-2-2-2-12-12-1222-1222-12222-12-12-2-2-2-2-2-2-2-2-1-12-1-1-1-12-1-1-1-12-1-1-1-1-1-122-1-1-1-1-1-1-1-122-1-1-1-1-1-12-1-11-21-21-21-21111-2-211111111-2-211111111-2-211111111-2-211110000000000002222-2-2-2-2000000000000000000000000-1-1-1-1111100000000
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χ1924-4-44-444-4-2-2-2000022-2-22-22-22-2-222222-2-22-22−232323−2323−23−2323111−1+3i32−1−3i32-1-1-11−3i321+3i32111−1+3i32−1−3i32-11-11-111−3i321+3i32−1+3i32−1−3i321-11-11-1−1+3i32−1−3i321−3i321+3i32-1-1-11−3i321+3i32000000003i−3i3−33−32ζ7+ζ11−ζ7−2ζ11−2ζ7−ζ11ζ7+2ζ11−3i3i−33−33−2ζ7−ζ11ζ7+2ζ112ζ7+ζ11−ζ7−2ζ11−3i3i−33−33−2ζ7−ζ11ζ7+2ζ112ζ7+ζ11−ζ7−2ζ113i−3i3−33−32ζ7+ζ11−ζ7−2ζ11−2ζ7−ζ11ζ7+2ζ11000000000000000000000000000000000000000000000000000000000000

Here ζ is a primitive root of unity of order 12 (ζ12 = 1); i = ζ3.