# Properties

 Modulus 8002 Structure $$C_{4000}$$ Order 4000

Show commands for: SageMath / Pari/GP

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

sage: H = DirichletGroup_conrey(8002)

pari: g = idealstar(,8002,2)

## Character group

 sage: G.order()  pari: g.no Order = 4000 sage: H.invariants()  pari: g.cyc Structure = $$C_{4000}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{8002}(3,\cdot)$

## First 32 of 4000 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 3 5 7 9 11 13 15 17 19 21
$$\chi_{8002}(1,\cdot)$$ 8002.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{8002}(3,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{1}{4000}\right)$$ $$e\left(\frac{59}{200}\right)$$ $$e\left(\frac{57}{1000}\right)$$ $$e\left(\frac{1}{2000}\right)$$ $$e\left(\frac{57}{160}\right)$$ $$e\left(\frac{541}{1000}\right)$$ $$e\left(\frac{1181}{4000}\right)$$ $$e\left(\frac{1431}{4000}\right)$$ $$e\left(\frac{981}{1000}\right)$$ $$e\left(\frac{229}{4000}\right)$$
$$\chi_{8002}(5,\cdot)$$ 8002.q 200 no $$1$$ $$1$$ $$e\left(\frac{59}{200}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{79}{200}\right)$$ $$e\left(\frac{29}{200}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{111}{200}\right)$$
$$\chi_{8002}(7,\cdot)$$ 8002.v 1000 no $$1$$ $$1$$ $$e\left(\frac{57}{1000}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{249}{250}\right)$$ $$e\left(\frac{57}{500}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{87}{250}\right)$$ $$e\left(\frac{317}{1000}\right)$$ $$e\left(\frac{567}{1000}\right)$$ $$e\left(\frac{167}{250}\right)$$ $$e\left(\frac{53}{1000}\right)$$
$$\chi_{8002}(9,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{1}{2000}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{57}{500}\right)$$ $$e\left(\frac{1}{1000}\right)$$ $$e\left(\frac{57}{80}\right)$$ $$e\left(\frac{41}{500}\right)$$ $$e\left(\frac{1181}{2000}\right)$$ $$e\left(\frac{1431}{2000}\right)$$ $$e\left(\frac{481}{500}\right)$$ $$e\left(\frac{229}{2000}\right)$$
$$\chi_{8002}(11,\cdot)$$ 8002.p 160 no $$-1$$ $$1$$ $$e\left(\frac{57}{160}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{57}{80}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{117}{160}\right)$$ $$e\left(\frac{127}{160}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{93}{160}\right)$$
$$\chi_{8002}(13,\cdot)$$ 8002.v 1000 no $$1$$ $$1$$ $$e\left(\frac{541}{1000}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{87}{250}\right)$$ $$e\left(\frac{41}{500}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{181}{250}\right)$$ $$e\left(\frac{921}{1000}\right)$$ $$e\left(\frac{171}{1000}\right)$$ $$e\left(\frac{221}{250}\right)$$ $$e\left(\frac{889}{1000}\right)$$
$$\chi_{8002}(15,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{1181}{4000}\right)$$ $$e\left(\frac{79}{200}\right)$$ $$e\left(\frac{317}{1000}\right)$$ $$e\left(\frac{1181}{2000}\right)$$ $$e\left(\frac{117}{160}\right)$$ $$e\left(\frac{921}{1000}\right)$$ $$e\left(\frac{2761}{4000}\right)$$ $$e\left(\frac{2011}{4000}\right)$$ $$e\left(\frac{561}{1000}\right)$$ $$e\left(\frac{2449}{4000}\right)$$
$$\chi_{8002}(17,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{1431}{4000}\right)$$ $$e\left(\frac{29}{200}\right)$$ $$e\left(\frac{567}{1000}\right)$$ $$e\left(\frac{1431}{2000}\right)$$ $$e\left(\frac{127}{160}\right)$$ $$e\left(\frac{171}{1000}\right)$$ $$e\left(\frac{2011}{4000}\right)$$ $$e\left(\frac{3761}{4000}\right)$$ $$e\left(\frac{811}{1000}\right)$$ $$e\left(\frac{3699}{4000}\right)$$
$$\chi_{8002}(19,\cdot)$$ 8002.v 1000 no $$1$$ $$1$$ $$e\left(\frac{981}{1000}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{167}{250}\right)$$ $$e\left(\frac{481}{500}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{221}{250}\right)$$ $$e\left(\frac{561}{1000}\right)$$ $$e\left(\frac{811}{1000}\right)$$ $$e\left(\frac{111}{250}\right)$$ $$e\left(\frac{649}{1000}\right)$$
$$\chi_{8002}(21,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{229}{4000}\right)$$ $$e\left(\frac{111}{200}\right)$$ $$e\left(\frac{53}{1000}\right)$$ $$e\left(\frac{229}{2000}\right)$$ $$e\left(\frac{93}{160}\right)$$ $$e\left(\frac{889}{1000}\right)$$ $$e\left(\frac{2449}{4000}\right)$$ $$e\left(\frac{3699}{4000}\right)$$ $$e\left(\frac{649}{1000}\right)$$ $$e\left(\frac{441}{4000}\right)$$
$$\chi_{8002}(23,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{3689}{4000}\right)$$ $$e\left(\frac{51}{200}\right)$$ $$e\left(\frac{273}{1000}\right)$$ $$e\left(\frac{1689}{2000}\right)$$ $$e\left(\frac{33}{160}\right)$$ $$e\left(\frac{749}{1000}\right)$$ $$e\left(\frac{709}{4000}\right)$$ $$e\left(\frac{2959}{4000}\right)$$ $$e\left(\frac{909}{1000}\right)$$ $$e\left(\frac{781}{4000}\right)$$
$$\chi_{8002}(25,\cdot)$$ 8002.n 100 no $$1$$ $$1$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$-i$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{29}{100}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{11}{100}\right)$$
$$\chi_{8002}(27,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{3}{4000}\right)$$ $$e\left(\frac{177}{200}\right)$$ $$e\left(\frac{171}{1000}\right)$$ $$e\left(\frac{3}{2000}\right)$$ $$e\left(\frac{11}{160}\right)$$ $$e\left(\frac{623}{1000}\right)$$ $$e\left(\frac{3543}{4000}\right)$$ $$e\left(\frac{293}{4000}\right)$$ $$e\left(\frac{943}{1000}\right)$$ $$e\left(\frac{687}{4000}\right)$$
$$\chi_{8002}(29,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{1109}{2000}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{213}{500}\right)$$ $$e\left(\frac{109}{1000}\right)$$ $$e\left(\frac{13}{80}\right)$$ $$e\left(\frac{469}{500}\right)$$ $$e\left(\frac{1729}{2000}\right)$$ $$e\left(\frac{979}{2000}\right)$$ $$e\left(\frac{429}{500}\right)$$ $$e\left(\frac{1961}{2000}\right)$$
$$\chi_{8002}(31,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{391}{2000}\right)$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{287}{500}\right)$$ $$e\left(\frac{391}{1000}\right)$$ $$e\left(\frac{47}{80}\right)$$ $$e\left(\frac{31}{500}\right)$$ $$e\left(\frac{1771}{2000}\right)$$ $$e\left(\frac{1521}{2000}\right)$$ $$e\left(\frac{71}{500}\right)$$ $$e\left(\frac{1539}{2000}\right)$$
$$\chi_{8002}(33,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{713}{2000}\right)$$ $$e\left(\frac{67}{100}\right)$$ $$e\left(\frac{141}{500}\right)$$ $$e\left(\frac{713}{1000}\right)$$ $$e\left(\frac{1}{80}\right)$$ $$e\left(\frac{233}{500}\right)$$ $$e\left(\frac{53}{2000}\right)$$ $$e\left(\frac{303}{2000}\right)$$ $$e\left(\frac{453}{500}\right)$$ $$e\left(\frac{1277}{2000}\right)$$
$$\chi_{8002}(35,\cdot)$$ 8002.o 125 no $$1$$ $$1$$ $$e\left(\frac{44}{125}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{32}{125}\right)$$ $$e\left(\frac{88}{125}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{91}{125}\right)$$ $$e\left(\frac{89}{125}\right)$$ $$e\left(\frac{89}{125}\right)$$ $$e\left(\frac{31}{125}\right)$$ $$e\left(\frac{76}{125}\right)$$
$$\chi_{8002}(37,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{2993}{4000}\right)$$ $$e\left(\frac{187}{200}\right)$$ $$e\left(\frac{601}{1000}\right)$$ $$e\left(\frac{993}{2000}\right)$$ $$e\left(\frac{41}{160}\right)$$ $$e\left(\frac{213}{1000}\right)$$ $$e\left(\frac{2733}{4000}\right)$$ $$e\left(\frac{2983}{4000}\right)$$ $$e\left(\frac{133}{1000}\right)$$ $$e\left(\frac{1397}{4000}\right)$$
$$\chi_{8002}(39,\cdot)$$ 8002.u 800 no $$-1$$ $$1$$ $$e\left(\frac{433}{800}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{81}{200}\right)$$ $$e\left(\frac{33}{400}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{53}{200}\right)$$ $$e\left(\frac{173}{800}\right)$$ $$e\left(\frac{423}{800}\right)$$ $$e\left(\frac{173}{200}\right)$$ $$e\left(\frac{757}{800}\right)$$
$$\chi_{8002}(41,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{2883}{4000}\right)$$ $$e\left(\frac{97}{200}\right)$$ $$e\left(\frac{331}{1000}\right)$$ $$e\left(\frac{883}{2000}\right)$$ $$e\left(\frac{11}{160}\right)$$ $$e\left(\frac{703}{1000}\right)$$ $$e\left(\frac{823}{4000}\right)$$ $$e\left(\frac{1573}{4000}\right)$$ $$e\left(\frac{223}{1000}\right)$$ $$e\left(\frac{207}{4000}\right)$$
$$\chi_{8002}(43,\cdot)$$ 8002.x 4000 no $$-1$$ $$1$$ $$e\left(\frac{3309}{4000}\right)$$ $$e\left(\frac{31}{200}\right)$$ $$e\left(\frac{613}{1000}\right)$$ $$e\left(\frac{1309}{2000}\right)$$ $$e\left(\frac{133}{160}\right)$$ $$e\left(\frac{169}{1000}\right)$$ $$e\left(\frac{3929}{4000}\right)$$ $$e\left(\frac{3179}{4000}\right)$$ $$e\left(\frac{129}{1000}\right)$$ $$e\left(\frac{1761}{4000}\right)$$
$$\chi_{8002}(45,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{591}{2000}\right)$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{187}{500}\right)$$ $$e\left(\frac{591}{1000}\right)$$ $$e\left(\frac{7}{80}\right)$$ $$e\left(\frac{231}{500}\right)$$ $$e\left(\frac{1971}{2000}\right)$$ $$e\left(\frac{1721}{2000}\right)$$ $$e\left(\frac{271}{500}\right)$$ $$e\left(\frac{1339}{2000}\right)$$
$$\chi_{8002}(47,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{741}{2000}\right)$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{237}{500}\right)$$ $$e\left(\frac{741}{1000}\right)$$ $$e\left(\frac{77}{80}\right)$$ $$e\left(\frac{381}{500}\right)$$ $$e\left(\frac{1121}{2000}\right)$$ $$e\left(\frac{371}{2000}\right)$$ $$e\left(\frac{421}{500}\right)$$ $$e\left(\frac{1689}{2000}\right)$$
$$\chi_{8002}(49,\cdot)$$ 8002.t 500 no $$1$$ $$1$$ $$e\left(\frac{57}{500}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{124}{125}\right)$$ $$e\left(\frac{57}{250}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{87}{125}\right)$$ $$e\left(\frac{317}{500}\right)$$ $$e\left(\frac{67}{500}\right)$$ $$e\left(\frac{42}{125}\right)$$ $$e\left(\frac{53}{500}\right)$$
$$\chi_{8002}(51,\cdot)$$ 8002.t 500 no $$1$$ $$1$$ $$e\left(\frac{179}{500}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{78}{125}\right)$$ $$e\left(\frac{179}{250}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{89}{125}\right)$$ $$e\left(\frac{399}{500}\right)$$ $$e\left(\frac{149}{500}\right)$$ $$e\left(\frac{99}{125}\right)$$ $$e\left(\frac{491}{500}\right)$$
$$\chi_{8002}(53,\cdot)$$ 8002.u 800 no $$-1$$ $$1$$ $$e\left(\frac{779}{800}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{3}{200}\right)$$ $$e\left(\frac{379}{400}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{39}{200}\right)$$ $$e\left(\frac{799}{800}\right)$$ $$e\left(\frac{349}{800}\right)$$ $$e\left(\frac{199}{200}\right)$$ $$e\left(\frac{791}{800}\right)$$
$$\chi_{8002}(55,\cdot)$$ 8002.u 800 no $$-1$$ $$1$$ $$e\left(\frac{521}{800}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{97}{200}\right)$$ $$e\left(\frac{121}{400}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{61}{200}\right)$$ $$e\left(\frac{101}{800}\right)$$ $$e\left(\frac{751}{800}\right)$$ $$e\left(\frac{101}{200}\right)$$ $$e\left(\frac{109}{800}\right)$$
$$\chi_{8002}(57,\cdot)$$ 8002.p 160 no $$-1$$ $$1$$ $$e\left(\frac{157}{160}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{77}{80}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{137}{160}\right)$$ $$e\left(\frac{27}{160}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{113}{160}\right)$$
$$\chi_{8002}(59,\cdot)$$ 8002.s 400 no $$1$$ $$1$$ $$e\left(\frac{381}{400}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{181}{200}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{361}{400}\right)$$ $$e\left(\frac{11}{400}\right)$$ $$e\left(\frac{61}{100}\right)$$ $$e\left(\frac{49}{400}\right)$$
$$\chi_{8002}(61,\cdot)$$ 8002.w 2000 no $$1$$ $$1$$ $$e\left(\frac{1421}{2000}\right)$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{497}{500}\right)$$ $$e\left(\frac{421}{1000}\right)$$ $$e\left(\frac{37}{80}\right)$$ $$e\left(\frac{261}{500}\right)$$ $$e\left(\frac{201}{2000}\right)$$ $$e\left(\frac{1451}{2000}\right)$$ $$e\left(\frac{1}{500}\right)$$ $$e\left(\frac{1409}{2000}\right)$$
$$\chi_{8002}(63,\cdot)$$ 8002.s 400 no $$1$$ $$1$$ $$e\left(\frac{23}{400}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{23}{200}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{363}{400}\right)$$ $$e\left(\frac{113}{400}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{67}{400}\right)$$