Finite Group Databasebeta

← Group #13047404

Character table — Group #13047404

Order 1296.

C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C23C24C25C26C27C28C29C30C31C32C33C34C35C36C37C38C39C40C41C42C43C44C45C46C47C48C49C50C51C52C53C54C55C56C57C58C59C60C61C62C63C64C65C66C67C68C69C70C71C72C73C74C75C76C77C78C79C80C81C82C83C84C85C86
size1122222666666666666666699991818181818181818181854545454545411222226666666666666666999918181818181818181818545454545454
order12236663333666666666666336666991818181818182466121222266666666666666666666666666181818181818181824661212
χ111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
χ21111111111111111111111111111111111111-1-1-1-1-1-11111111111111111111111111111111111111-1-1-1-1-1-1
χ311111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i3211−1+i32−1−i32−1+i32−1−i3211111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i3211−1+i32−1−i32−1+i32−1−i32
χ411111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i32-1-11−i321+i321−i321+i3211111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i32-1-11−i321+i321−i321+i32
χ511111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i3211−1−i32−1+i32−1−i32−1+i3211111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i3211−1−i32−1+i32−1−i32−1+i32
χ611111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i32-1-11+i321−i321+i321−i3211111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i32-1-11+i321−i321+i321−i32
χ711-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-11-111-1-111-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-11-111-1-1
χ811-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-1-11-1-11111-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-1-11-1-111
χ911-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i321-1−1+i32−1−i321−i321+i3211-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i321-1−1+i32−1−i321−i321+i32
χ1011-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i32-111−i321+i32−1+i32−1−i3211-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i32-111−i321+i32−1+i32−1−i32
χ1111-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i321-1−1−i32−1+i321+i321−i3211-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i321-1−1−i32−1+i321+i321−i32
χ1211-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i32-111+i321−i32−1−i32−1+i3211-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i32-111+i321−i32−1−i32−1+i32
χ1322222222-1-1-12-1-1-122-1-1-1-1-1-1222222-1-1-1-1-1-1-1-100000022222222-1-1-12-1-1-122-1-1-1-1-1-1222222-1-1-1-1-1-1-1-1000000
χ1422222222-1-1-12-1-1-122-1-1-1-1-1-1−1+i3−1−i3−1+i3−1−i3−1+i3−1−i31−i321+i321−i321−i321−i321+i321+i321+i3200000022222222-1-1-12-1-1-122-1-1-1-1-1-1−1+i3−1−i3−1+i3−1−i3−1+i3−1−i31−i321+i321−i321−i321−i321+i321+i321+i32000000
χ1522222222-1-1-12-1-1-122-1-1-1-1-1-1−1−i3−1+i3−1−i3−1+i3−1−i3−1+i31+i321−i321+i321+i321+i321−i321−i321−i3200000022222222-1-1-12-1-1-122-1-1-1-1-1-1−1−i3−1+i3−1−i3−1+i3−1−i3−1+i31+i321−i321+i321+i321+i321−i321−i321−i32000000
χ1622-222-2-22-1-1-12-1-1-1-2-21111112222-2-2-1-1-111-11100000022-222-2-22-1-1-12-1-1-1-2-21111112222-2-2-1-1-111-111000000
χ1722-222-2-22-1-1-12-1-1-1-2-2111111−1+i3−1−i3−1+i3−1−i31−i31+i31−i321+i321−i32−1+i32−1+i321+i32−1−i32−1−i3200000022-222-2-22-1-1-12-1-1-1-2-2111111−1+i3−1−i3−1+i3−1−i31−i31+i31−i321+i321−i32−1+i32−1+i321+i32−1−i32−1−i32000000
χ1822-222-2-22-1-1-12-1-1-1-2-2111111−1−i3−1+i3−1−i3−1+i31+i31−i31+i321−i321+i32−1−i32−1−i321−i32−1+i32−1+i3200000022-222-2-22-1-1-12-1-1-1-2-2111111−1−i3−1+i3−1−i3−1+i31+i31−i31+i321−i321+i32−1−i32−1−i321−i32−1+i32−1+i32000000
χ192-202-2002222-2-2-2-20000000022-2-20022-200-2000000002-202-2002222-2-2-2-20000000022-2-20022-200-200000000
χ202-202-2002222-2-2-2-200000000−1+i3−1−i31−i31+i300−1+i3−1−i31−i3001+i3000000002-202-2002222-2-2-2-200000000−1+i3−1−i31−i31+i300−1+i3−1−i31−i3001+i300000000
χ212-202-2002222-2-2-2-200000000−1−i3−1+i31+i31−i300−1−i3−1+i31+i3001−i3000000002-202-2002222-2-2-2-200000000−1−i3−1+i31+i31−i300−1−i3−1+i31+i3001−i300000000
χ222-202-2002-1-1-1-211100i3i3i3−i3−i3−i322-2-200-1-11i3−i31i3−i30000002-202-2002-1-1-1-211100i3i3i3−i3−i3−i322-2-200-1-11i3−i31i3−i3000000
χ232-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1+i3−1−i31−i31+i3001−i321+i32−1+i32−3−i323+i32−1−i323−i32−3+i320000002-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1+i3−1−i31−i31+i3001−i321+i32−1+i32−3−i323+i32−1−i323−i32−3+i32000000
χ242-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1−i3−1+i31+i31−i3001+i321−i32−1−i323−i32−3+i32−1+i32−3−i323+i320000002-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1−i3−1+i31+i31−i3001+i321−i32−1−i323−i32−3+i32−1+i32−3−i323+i32000000
χ252-202-2002-1-1-1-211100−i3−i3−i3i3i3i322-2-200-1-11−i3i31−i3i30000002-202-2002-1-1-1-211100−i3−i3−i3i3i3i322-2-200-1-11−i3i31−i3i3000000
χ262-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1+i3−1−i31−i31+i3001−i321+i32−1+i323+i32−3−i32−1−i32−3+i323−i320000002-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1+i3−1−i31−i31+i3001−i321+i32−1+i323+i32−3−i32−1−i32−3+i323−i32000000
χ272-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1−i3−1+i31+i31−i3001+i321−i32−1−i32−3+i323−i32−1+i323+i32−3−i320000002-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1−i3−1+i31+i31−i3001+i321−i32−1−i32−3+i323−i32−1+i323+i32−3−i32000000
χ286666666-3000-3000-3-3000000000000000000000000006666666-3000-3000-3-300000000000000000000000000
χ29666-3-3-3-3003-3003-30003-303-300000000000000000000666-3-3-3-3003-3003-30003-303-300000000000000000000
χ30666-3-3-3-303-3003-30003-303-3000000000000000000000666-3-3-3-303-3003-30003-303-3000000000000000000000
χ31666-3-3-3-30-3030-30300-303-30300000000000000000000666-3-3-3-30-3030-30300-303-30300000000000000000000
χ3266-666-6-6-3000-3000330000000000000000000000000066-666-6-6-3000-30003300000000000000000000000000
χ3366-6-3-333003-3003-3000-330-330000000000000000000066-6-3-333003-3003-3000-330-3300000000000000000000
χ3466-6-3-33303-3003-3000-330-3300000000000000000000066-6-3-33303-3003-3000-330-33000000000000000000000
χ3566-6-3-3330-3030-3030030-330-30000000000000000000066-6-3-3330-3030-3030030-330-300000000000000000000
χ366-606-600-300030003i3−3i3000000000000000000000000006-606-600-300030003i3−3i300000000000000000000000000
χ376-606-600-30003000−3i33i3000000000000000000000000006-606-600-30003000−3i33i300000000000000000000000000
χ386-60-333i3−3i3003-300-33002i3−i3−i3−2i3i3i3000000000000000000006-60-333i3−3i3003-300-33002i3−i3−i3−2i3i3i300000000000000000000
χ396-60-333i3−3i303-300-33000−i3−i32i3i3i3−2i3000000000000000000006-60-333i3−3i303-300-33000−i3−i32i3i3i3−2i300000000000000000000
χ406-60-333i3−3i30-303030-300−i32i3−i3i3−2i3i3000000000000000000006-60-333i3−3i30-303030-300−i32i3−i3i3−2i3i300000000000000000000
χ416-60-33−3i33i3003-300-3300−2i3i3i32i3−i3−i3000000000000000000006-60-33−3i33i3003-300-3300−2i3i3i32i3−i3−i300000000000000000000
χ426-60-33−3i33i303-300-33000i3i3−2i3−i3−i32i3000000000000000000006-60-33−3i33i303-300-33000i3i3−2i3−i3−i32i300000000000000000000
χ436-60-33−3i33i30-303030-300i3−2i3i3−i32i3−i3000000000000000000006-60-33−3i33i30-303030-300i3−2i3i3−i32i3−i300000000000000000000
χ441111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
χ451111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111
χ4611111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i3211−1+i32−1−i32−1+i32−1−i32-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11−i321+i321−i321+i321−i321+i321−i321+i321−i321−i321−i321+i321+i321+i32-1-11−i321+i321−i321+i32
χ4711111111111111111111111−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i32-1-11−i321+i321−i321+i32-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11−i321+i321−i321+i321−i321+i321−i321+i321−i321−i321−i321+i321+i321+i3211−1+i32−1−i32−1+i32−1−i32
χ4811111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i3211−1−i32−1+i32−1−i32−1+i32-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11+i321−i321+i321−i321+i321−i321+i321−i321+i321+i321+i321−i321−i321−i32-1-11+i321−i321+i321−i32
χ4911111111111111111111111−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i32-1-11+i321−i321+i321−i32-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11+i321−i321+i321−i321+i321−i321+i321−i321+i321+i321+i321−i321−i321−i3211−1−i32−1+i32−1−i32−1+i32
χ5011-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-11-111-1-1-1-11-1-111-1-1-1-1-1-1-1-111111111-1-1-1-111-1-1-111-111-11-1-111
χ5111-111-1-111111111-1-1-1-1-1-1-1-11111-1-1111-1-11-1-1-11-1-111-1-11-1-111-1-1-1-1-1-1-1-111111111-1-1-1-111-1-1-111-1111-111-1-1
χ5211-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i321-1−1+i32−1−i321−i321+i32-1-11-1-111-1-1-1-1-1-1-1-1111111111−i321+i321−i321+i32−1+i32−1−i321−i321+i321−i32−1+i32−1+i321+i32−1−i32−1−i32-111−i321+i32−1+i32−1−i32
χ5311-111-1-111111111-1-1-1-1-1-1-1-1−1+i32−1−i32−1+i32−1−i321−i321+i32−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i32-111−i321+i32−1+i32−1−i32-1-11-1-111-1-1-1-1-1-1-1-1111111111−i321+i321−i321+i32−1+i32−1−i321−i321+i321−i32−1+i32−1+i321+i32−1−i32−1−i321-1−1+i32−1−i321−i321+i32
χ5411-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i321-1−1−i32−1+i321+i321−i32-1-11-1-111-1-1-1-1-1-1-1-1111111111+i321−i321+i321−i32−1−i32−1+i321+i321−i321+i32−1−i32−1−i321−i32−1+i32−1+i32-111+i321−i32−1−i32−1+i32
χ5511-111-1-111111111-1-1-1-1-1-1-1-1−1−i32−1+i32−1−i32−1+i321+i321−i32−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i32-111+i321−i32−1−i32−1+i32-1-11-1-111-1-1-1-1-1-1-1-1111111111+i321−i321+i321−i32−1−i32−1+i321+i321−i321+i32−1−i32−1−i321−i32−1+i32−1+i321-1−1−i32−1+i321+i321−i32
χ5622222222-1-1-12-1-1-122-1-1-1-1-1-1222222-1-1-1-1-1-1-1-1000000-2-2-2-2-2-2-2-2111-2111-2-2111111-2-2-2-2-2-211111111000000
χ5722222222-1-1-12-1-1-122-1-1-1-1-1-1−1+i3−1−i3−1+i3−1−i3−1+i3−1−i31−i321+i321−i321−i321−i321+i321+i321+i32000000-2-2-2-2-2-2-2-2111-2111-2-21111111−i31+i31−i31+i31−i31+i3−1+i32−1−i32−1+i32−1+i32−1+i32−1−i32−1−i32−1−i32000000
χ5822222222-1-1-12-1-1-122-1-1-1-1-1-1−1−i3−1+i3−1−i3−1+i3−1−i3−1+i31+i321−i321+i321+i321+i321−i321−i321−i32000000-2-2-2-2-2-2-2-2111-2111-2-21111111+i31−i31+i31−i31+i31−i3−1−i32−1+i32−1−i32−1−i32−1−i32−1+i32−1+i32−1+i32000000
χ5922-222-2-22-1-1-12-1-1-1-2-21111112222-2-2-1-1-111-111000000-2-22-2-222-2111-211122-1-1-1-1-1-1-2-2-2-222111-1-11-1-1000000
χ6022-222-2-22-1-1-12-1-1-1-2-2111111−1+i3−1−i3−1+i3−1−i31−i31+i31−i321+i321−i32−1+i32−1+i321+i32−1−i32−1−i32000000-2-22-2-222-2111-211122-1-1-1-1-1-11−i31+i31−i31+i3−1+i3−1−i3−1+i32−1−i32−1+i321−i321−i32−1−i321+i321+i32000000
χ6122-222-2-22-1-1-12-1-1-1-2-2111111−1−i3−1+i3−1−i3−1+i31+i31−i31+i321−i321+i32−1−i32−1−i321−i32−1+i32−1+i32000000-2-22-2-222-2111-211122-1-1-1-1-1-11+i31−i31+i31−i3−1−i3−1+i3−1−i32−1+i32−1−i321+i321+i32−1+i321−i321−i32000000
χ622-202-2002222-2-2-2-20000000022-2-20022-200-200000000-220-2200-2-2-2-2222200000000-2-22200-2-2200200000000
χ632-202-2002222-2-2-2-200000000−1+i3−1−i31−i31+i300−1+i3−1−i31−i3001+i300000000-220-2200-2-2-2-22222000000001−i31+i3−1+i3−1−i3001−i31+i3−1+i300−1−i300000000
χ642-202-2002222-2-2-2-200000000−1−i3−1+i31+i31−i300−1−i3−1+i31+i3001−i300000000-220-2200-2-2-2-22222000000001+i31−i3−1−i3−1+i3001+i31−i3−1−i300−1+i300000000
χ652-202-2002-1-1-1-211100i3i3i3−i3−i3−i322-2-200-1-11i3−i31i3−i3000000-220-2200-21112-1-1-100−i3−i3−i3i3i3i3-2-2220011-1−i3i3-1−i3i3000000
χ662-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1+i3−1−i31−i31+i3001−i321+i32−1+i32−3−i323+i32−1−i323−i32−3+i32000000-220-2200-21112-1-1-100−i3−i3−i3i3i3i31−i31+i3−1+i3−1−i300−1+i32−1−i321−i323+i32−3−i321+i32−3+i323−i32000000
χ672-202-2002-1-1-1-211100i3i3i3−i3−i3−i3−1−i3−1+i31+i31−i3001+i321−i32−1−i323−i32−3+i32−1+i32−3−i323+i32000000-220-2200-21112-1-1-100−i3−i3−i3i3i3i31+i31−i3−1−i3−1+i300−1−i32−1+i321+i32−3+i323−i321−i323+i32−3−i32000000
χ682-202-2002-1-1-1-211100−i3−i3−i3i3i3i322-2-200-1-11−i3i31−i3i3000000-220-2200-21112-1-1-100i3i3i3−i3−i3−i3-2-2220011-1i3−i3-1i3−i3000000
χ692-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1+i3−1−i31−i31+i3001−i321+i32−1+i323+i32−3−i32−1−i32−3+i323−i32000000-220-2200-21112-1-1-100i3i3i3−i3−i3−i31−i31+i3−1+i3−1−i300−1+i32−1−i321−i32−3−i323+i321+i323−i32−3+i32000000
χ702-202-2002-1-1-1-211100−i3−i3−i3i3i3i3−1−i3−1+i31+i31−i3001+i321−i32−1−i32−3+i323−i32−1+i323+i32−3−i32000000-220-2200-21112-1-1-100i3i3i3−i3−i3−i31+i31−i3−1−i3−1+i300−1−i32−1+i321+i323−i32−3+i321−i32−3−i323+i32000000
χ716666666-3000-3000-3-300000000000000000000000000-6-6-6-6-6-6-6300030003300000000000000000000000000
χ72666-3-3-3-3003-3003-30003-303-300000000000000000000-6-6-6333300-3300-33000-330-3300000000000000000000
χ73666-3-3-3-303-3003-30003-303-3000000000000000000000-6-6-633330-3300-33000-330-33000000000000000000000
χ74666-3-3-3-30-3030-30300-303-30300000000000000000000-6-6-63333030-3030-30030-330-300000000000000000000
χ7566-666-6-6-3000-30003300000000000000000000000000-6-66-6-66630003000-3-300000000000000000000000000
χ7666-6-3-333003-3003-3000-330-3300000000000000000000-6-6633-3-300-3300-330003-303-300000000000000000000
χ7766-6-3-33303-3003-3000-330-33000000000000000000000-6-6633-3-30-3300-330003-303-3000000000000000000000
χ7866-6-3-3330-3030-3030030-330-300000000000000000000-6-6633-3-3030-3030-300-303-30300000000000000000000
χ796-606-600-300030003i3−3i300000000000000000000000000-660-66003000-3000−3i33i300000000000000000000000000
χ806-606-600-30003000−3i33i300000000000000000000000000-660-66003000-30003i3−3i300000000000000000000000000
χ816-60-333i3−3i3003-300-33002i3−i3−i3−2i3i3i300000000000000000000-6603-3−3i33i300-33003-300−2i3i3i32i3−i3−i300000000000000000000
χ826-60-333i3−3i303-300-33000−i3−i32i3i3i3−2i300000000000000000000-6603-3−3i33i30-33003-3000i3i3−2i3−i3−i32i300000000000000000000
χ836-60-333i3−3i30-303030-300−i32i3−i3i3−2i3i300000000000000000000-6603-3−3i33i3030-30-30300i3−2i3i3−i32i3−i300000000000000000000
χ846-60-33−3i33i3003-300-3300−2i3i3i32i3−i3−i300000000000000000000-6603-33i3−3i300-33003-3002i3−i3−i3−2i3i3i300000000000000000000
χ856-60-33−3i33i303-300-33000i3i3−2i3−i3−i32i300000000000000000000-6603-33i3−3i30-33003-3000−i3−i32i3i3i3−2i300000000000000000000
χ866-60-33−3i33i30-303030-300i3−2i3i3−i32i3−i300000000000000000000-6603-33i3−3i3030-30-30300−i32i3−i3i3−2i3i300000000000000000000

Here i is the imaginary unit (i² = −1).