# Properties

 Modulus 4003 Structure $$C_{4002}$$ Order 4002

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(4003)

pari: g = idealstar(,4003,2)

## Character group

 sage: G.order()  pari: g.no Order = 4002 sage: H.invariants()  pari: g.cyc Structure = $$C_{4002}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{4003}(2,\cdot)$

## First 32 of 4002 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{4003}(1,\cdot)$$ 4003.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{4003}(2,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{4002}\right)$$ $$e\left(\frac{2417}{4002}\right)$$ $$e\left(\frac{1}{2001}\right)$$ $$e\left(\frac{1471}{4002}\right)$$ $$e\left(\frac{403}{667}\right)$$ $$e\left(\frac{1067}{2001}\right)$$ $$e\left(\frac{1}{1334}\right)$$ $$e\left(\frac{416}{2001}\right)$$ $$e\left(\frac{32}{87}\right)$$ $$e\left(\frac{281}{667}\right)$$
$$\chi_{4003}(3,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{2417}{4002}\right)$$ $$e\left(\frac{2971}{4002}\right)$$ $$e\left(\frac{416}{2001}\right)$$ $$e\left(\frac{1631}{4002}\right)$$ $$e\left(\frac{231}{667}\right)$$ $$e\left(\frac{1651}{2001}\right)$$ $$e\left(\frac{1083}{1334}\right)$$ $$e\left(\frac{970}{2001}\right)$$ $$e\left(\frac{1}{87}\right)$$ $$e\left(\frac{171}{667}\right)$$
$$\chi_{4003}(4,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{1}{2001}\right)$$ $$e\left(\frac{416}{2001}\right)$$ $$e\left(\frac{2}{2001}\right)$$ $$e\left(\frac{1471}{2001}\right)$$ $$e\left(\frac{139}{667}\right)$$ $$e\left(\frac{133}{2001}\right)$$ $$e\left(\frac{1}{667}\right)$$ $$e\left(\frac{832}{2001}\right)$$ $$e\left(\frac{64}{87}\right)$$ $$e\left(\frac{562}{667}\right)$$
$$\chi_{4003}(5,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{1471}{4002}\right)$$ $$e\left(\frac{1631}{4002}\right)$$ $$e\left(\frac{1471}{2001}\right)$$ $$e\left(\frac{2761}{4002}\right)$$ $$e\left(\frac{517}{667}\right)$$ $$e\left(\frac{773}{2001}\right)$$ $$e\left(\frac{137}{1334}\right)$$ $$e\left(\frac{1631}{2001}\right)$$ $$e\left(\frac{5}{87}\right)$$ $$e\left(\frac{478}{667}\right)$$
$$\chi_{4003}(6,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{403}{667}\right)$$ $$e\left(\frac{231}{667}\right)$$ $$e\left(\frac{139}{667}\right)$$ $$e\left(\frac{517}{667}\right)$$ $$e\left(\frac{634}{667}\right)$$ $$e\left(\frac{239}{667}\right)$$ $$e\left(\frac{542}{667}\right)$$ $$e\left(\frac{462}{667}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{452}{667}\right)$$
$$\chi_{4003}(7,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{1067}{2001}\right)$$ $$e\left(\frac{1651}{2001}\right)$$ $$e\left(\frac{133}{2001}\right)$$ $$e\left(\frac{773}{2001}\right)$$ $$e\left(\frac{239}{667}\right)$$ $$e\left(\frac{1841}{2001}\right)$$ $$e\left(\frac{400}{667}\right)$$ $$e\left(\frac{1301}{2001}\right)$$ $$e\left(\frac{80}{87}\right)$$ $$e\left(\frac{21}{667}\right)$$
$$\chi_{4003}(8,\cdot)$$ 4003.n 1334 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{1334}\right)$$ $$e\left(\frac{1083}{1334}\right)$$ $$e\left(\frac{1}{667}\right)$$ $$e\left(\frac{137}{1334}\right)$$ $$e\left(\frac{542}{667}\right)$$ $$e\left(\frac{400}{667}\right)$$ $$e\left(\frac{3}{1334}\right)$$ $$e\left(\frac{416}{667}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{176}{667}\right)$$
$$\chi_{4003}(9,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{416}{2001}\right)$$ $$e\left(\frac{970}{2001}\right)$$ $$e\left(\frac{832}{2001}\right)$$ $$e\left(\frac{1631}{2001}\right)$$ $$e\left(\frac{462}{667}\right)$$ $$e\left(\frac{1301}{2001}\right)$$ $$e\left(\frac{416}{667}\right)$$ $$e\left(\frac{1940}{2001}\right)$$ $$e\left(\frac{2}{87}\right)$$ $$e\left(\frac{342}{667}\right)$$
$$\chi_{4003}(10,\cdot)$$ 4003.j 87 Yes $$1$$ $$1$$ $$e\left(\frac{32}{87}\right)$$ $$e\left(\frac{1}{87}\right)$$ $$e\left(\frac{64}{87}\right)$$ $$e\left(\frac{5}{87}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{80}{87}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{2}{87}\right)$$ $$e\left(\frac{37}{87}\right)$$ $$e\left(\frac{4}{29}\right)$$
$$\chi_{4003}(11,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{281}{667}\right)$$ $$e\left(\frac{171}{667}\right)$$ $$e\left(\frac{562}{667}\right)$$ $$e\left(\frac{478}{667}\right)$$ $$e\left(\frac{452}{667}\right)$$ $$e\left(\frac{21}{667}\right)$$ $$e\left(\frac{176}{667}\right)$$ $$e\left(\frac{342}{667}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{196}{667}\right)$$
$$\chi_{4003}(12,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{2419}{4002}\right)$$ $$e\left(\frac{3803}{4002}\right)$$ $$e\left(\frac{418}{2001}\right)$$ $$e\left(\frac{571}{4002}\right)$$ $$e\left(\frac{370}{667}\right)$$ $$e\left(\frac{1784}{2001}\right)$$ $$e\left(\frac{1085}{1334}\right)$$ $$e\left(\frac{1802}{2001}\right)$$ $$e\left(\frac{65}{87}\right)$$ $$e\left(\frac{66}{667}\right)$$
$$\chi_{4003}(13,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{30}{667}\right)$$ $$e\left(\frac{474}{667}\right)$$ $$e\left(\frac{60}{667}\right)$$ $$e\left(\frac{108}{667}\right)$$ $$e\left(\frac{504}{667}\right)$$ $$e\left(\frac{655}{667}\right)$$ $$e\left(\frac{90}{667}\right)$$ $$e\left(\frac{281}{667}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{555}{667}\right)$$
$$\chi_{4003}(14,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{2135}{4002}\right)$$ $$e\left(\frac{1717}{4002}\right)$$ $$e\left(\frac{134}{2001}\right)$$ $$e\left(\frac{3017}{4002}\right)$$ $$e\left(\frac{642}{667}\right)$$ $$e\left(\frac{907}{2001}\right)$$ $$e\left(\frac{801}{1334}\right)$$ $$e\left(\frac{1717}{2001}\right)$$ $$e\left(\frac{25}{87}\right)$$ $$e\left(\frac{302}{667}\right)$$
$$\chi_{4003}(15,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{648}{667}\right)$$ $$e\left(\frac{100}{667}\right)$$ $$e\left(\frac{629}{667}\right)$$ $$e\left(\frac{65}{667}\right)$$ $$e\left(\frac{81}{667}\right)$$ $$e\left(\frac{141}{667}\right)$$ $$e\left(\frac{610}{667}\right)$$ $$e\left(\frac{200}{667}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{649}{667}\right)$$
$$\chi_{4003}(16,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{2}{2001}\right)$$ $$e\left(\frac{832}{2001}\right)$$ $$e\left(\frac{4}{2001}\right)$$ $$e\left(\frac{941}{2001}\right)$$ $$e\left(\frac{278}{667}\right)$$ $$e\left(\frac{266}{2001}\right)$$ $$e\left(\frac{2}{667}\right)$$ $$e\left(\frac{1664}{2001}\right)$$ $$e\left(\frac{41}{87}\right)$$ $$e\left(\frac{457}{667}\right)$$
$$\chi_{4003}(17,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{655}{667}\right)$$ $$e\left(\frac{344}{667}\right)$$ $$e\left(\frac{643}{667}\right)$$ $$e\left(\frac{357}{667}\right)$$ $$e\left(\frac{332}{667}\right)$$ $$e\left(\frac{405}{667}\right)$$ $$e\left(\frac{631}{667}\right)$$ $$e\left(\frac{21}{667}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{445}{667}\right)$$
$$\chi_{4003}(18,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{833}{4002}\right)$$ $$e\left(\frac{355}{4002}\right)$$ $$e\left(\frac{833}{2001}\right)$$ $$e\left(\frac{731}{4002}\right)$$ $$e\left(\frac{198}{667}\right)$$ $$e\left(\frac{367}{2001}\right)$$ $$e\left(\frac{833}{1334}\right)$$ $$e\left(\frac{355}{2001}\right)$$ $$e\left(\frac{34}{87}\right)$$ $$e\left(\frac{623}{667}\right)$$
$$\chi_{4003}(19,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{80}{2001}\right)$$ $$e\left(\frac{1264}{2001}\right)$$ $$e\left(\frac{160}{2001}\right)$$ $$e\left(\frac{1622}{2001}\right)$$ $$e\left(\frac{448}{667}\right)$$ $$e\left(\frac{635}{2001}\right)$$ $$e\left(\frac{80}{667}\right)$$ $$e\left(\frac{527}{2001}\right)$$ $$e\left(\frac{74}{87}\right)$$ $$e\left(\frac{271}{667}\right)$$
$$\chi_{4003}(20,\cdot)$$ 4003.n 1334 Yes $$-1$$ $$1$$ $$e\left(\frac{491}{1334}\right)$$ $$e\left(\frac{821}{1334}\right)$$ $$e\left(\frac{491}{667}\right)$$ $$e\left(\frac{567}{1334}\right)$$ $$e\left(\frac{656}{667}\right)$$ $$e\left(\frac{302}{667}\right)$$ $$e\left(\frac{139}{1334}\right)$$ $$e\left(\frac{154}{667}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{373}{667}\right)$$
$$\chi_{4003}(21,\cdot)$$ 4003.n 1334 Yes $$-1$$ $$1$$ $$e\left(\frac{183}{1334}\right)$$ $$e\left(\frac{757}{1334}\right)$$ $$e\left(\frac{183}{667}\right)$$ $$e\left(\frac{1059}{1334}\right)$$ $$e\left(\frac{470}{667}\right)$$ $$e\left(\frac{497}{667}\right)$$ $$e\left(\frac{549}{1334}\right)$$ $$e\left(\frac{90}{667}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{192}{667}\right)$$
$$\chi_{4003}(22,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{1687}{4002}\right)$$ $$e\left(\frac{3443}{4002}\right)$$ $$e\left(\frac{1687}{2001}\right)$$ $$e\left(\frac{337}{4002}\right)$$ $$e\left(\frac{188}{667}\right)$$ $$e\left(\frac{1130}{2001}\right)$$ $$e\left(\frac{353}{1334}\right)$$ $$e\left(\frac{1442}{2001}\right)$$ $$e\left(\frac{44}{87}\right)$$ $$e\left(\frac{477}{667}\right)$$
$$\chi_{4003}(23,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{613}{4002}\right)$$ $$e\left(\frac{881}{4002}\right)$$ $$e\left(\frac{613}{2001}\right)$$ $$e\left(\frac{1273}{4002}\right)$$ $$e\left(\frac{249}{667}\right)$$ $$e\left(\frac{1745}{2001}\right)$$ $$e\left(\frac{613}{1334}\right)$$ $$e\left(\frac{881}{2001}\right)$$ $$e\left(\frac{41}{87}\right)$$ $$e\left(\frac{167}{667}\right)$$
$$\chi_{4003}(24,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{1210}{2001}\right)$$ $$e\left(\frac{1109}{2001}\right)$$ $$e\left(\frac{419}{2001}\right)$$ $$e\left(\frac{1021}{2001}\right)$$ $$e\left(\frac{106}{667}\right)$$ $$e\left(\frac{850}{2001}\right)$$ $$e\left(\frac{543}{667}\right)$$ $$e\left(\frac{217}{2001}\right)$$ $$e\left(\frac{10}{87}\right)$$ $$e\left(\frac{347}{667}\right)$$
$$\chi_{4003}(25,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{1471}{2001}\right)$$ $$e\left(\frac{1631}{2001}\right)$$ $$e\left(\frac{941}{2001}\right)$$ $$e\left(\frac{760}{2001}\right)$$ $$e\left(\frac{367}{667}\right)$$ $$e\left(\frac{1546}{2001}\right)$$ $$e\left(\frac{137}{667}\right)$$ $$e\left(\frac{1261}{2001}\right)$$ $$e\left(\frac{10}{87}\right)$$ $$e\left(\frac{289}{667}\right)$$
$$\chi_{4003}(26,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{181}{4002}\right)$$ $$e\left(\frac{1259}{4002}\right)$$ $$e\left(\frac{181}{2001}\right)$$ $$e\left(\frac{2119}{4002}\right)$$ $$e\left(\frac{240}{667}\right)$$ $$e\left(\frac{1031}{2001}\right)$$ $$e\left(\frac{181}{1334}\right)$$ $$e\left(\frac{1259}{2001}\right)$$ $$e\left(\frac{50}{87}\right)$$ $$e\left(\frac{169}{667}\right)$$
$$\chi_{4003}(27,\cdot)$$ 4003.n 1334 Yes $$-1$$ $$1$$ $$e\left(\frac{1083}{1334}\right)$$ $$e\left(\frac{303}{1334}\right)$$ $$e\left(\frac{416}{667}\right)$$ $$e\left(\frac{297}{1334}\right)$$ $$e\left(\frac{26}{667}\right)$$ $$e\left(\frac{317}{667}\right)$$ $$e\left(\frac{581}{1334}\right)$$ $$e\left(\frac{303}{667}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{513}{667}\right)$$
$$\chi_{4003}(28,\cdot)$$ 4003.m 667 Yes $$1$$ $$1$$ $$e\left(\frac{356}{667}\right)$$ $$e\left(\frac{22}{667}\right)$$ $$e\left(\frac{45}{667}\right)$$ $$e\left(\frac{81}{667}\right)$$ $$e\left(\frac{378}{667}\right)$$ $$e\left(\frac{658}{667}\right)$$ $$e\left(\frac{401}{667}\right)$$ $$e\left(\frac{44}{667}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{583}{667}\right)$$
$$\chi_{4003}(29,\cdot)$$ 4003.o 2001 Yes $$1$$ $$1$$ $$e\left(\frac{1486}{2001}\right)$$ $$e\left(\frac{1868}{2001}\right)$$ $$e\left(\frac{971}{2001}\right)$$ $$e\left(\frac{814}{2001}\right)$$ $$e\left(\frac{451}{667}\right)$$ $$e\left(\frac{1540}{2001}\right)$$ $$e\left(\frac{152}{667}\right)$$ $$e\left(\frac{1735}{2001}\right)$$ $$e\left(\frac{13}{87}\right)$$ $$e\left(\frac{48}{667}\right)$$
$$\chi_{4003}(30,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{3889}{4002}\right)$$ $$e\left(\frac{3017}{4002}\right)$$ $$e\left(\frac{1888}{2001}\right)$$ $$e\left(\frac{1861}{4002}\right)$$ $$e\left(\frac{484}{667}\right)$$ $$e\left(\frac{1490}{2001}\right)$$ $$e\left(\frac{1221}{1334}\right)$$ $$e\left(\frac{1016}{2001}\right)$$ $$e\left(\frac{38}{87}\right)$$ $$e\left(\frac{263}{667}\right)$$
$$\chi_{4003}(31,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{1597}{4002}\right)$$ $$e\left(\frac{2021}{4002}\right)$$ $$e\left(\frac{1597}{2001}\right)$$ $$e\left(\frac{13}{4002}\right)$$ $$e\left(\frac{603}{667}\right)$$ $$e\left(\frac{1148}{2001}\right)$$ $$e\left(\frac{263}{1334}\right)$$ $$e\left(\frac{20}{2001}\right)$$ $$e\left(\frac{35}{87}\right)$$ $$e\left(\frac{533}{667}\right)$$
$$\chi_{4003}(32,\cdot)$$ 4003.p 4002 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{4002}\right)$$ $$e\left(\frac{79}{4002}\right)$$ $$e\left(\frac{5}{2001}\right)$$ $$e\left(\frac{3353}{4002}\right)$$ $$e\left(\frac{14}{667}\right)$$ $$e\left(\frac{1333}{2001}\right)$$ $$e\left(\frac{5}{1334}\right)$$ $$e\left(\frac{79}{2001}\right)$$ $$e\left(\frac{73}{87}\right)$$ $$e\left(\frac{71}{667}\right)$$