# Properties

 Label 2001.bu Modulus $2001$ Conductor $667$ Order $308$ Real no Primitive no Minimal yes Parity even

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(2001, base_ring=CyclotomicField(308))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,42,253]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(10,2001))

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$2001$$ Conductor: $$667$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$308$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 667.x sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{308})$ Fixed field: Number field defined by a degree 308 polynomial (not computed)

## First 31 of 120 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{2001}(10,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{308}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{87}{308}\right)$$ $$e\left(\frac{93}{308}\right)$$ $$e\left(\frac{235}{308}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{167}{308}\right)$$ $$e\left(\frac{29}{77}\right)$$
$$\chi_{2001}(19,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{211}{308}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{17}{308}\right)$$ $$e\left(\frac{135}{308}\right)$$ $$e\left(\frac{53}{308}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{153}{308}\right)$$ $$e\left(\frac{57}{77}\right)$$
$$\chi_{2001}(37,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{308}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{15}{308}\right)$$ $$e\left(\frac{101}{308}\right)$$ $$e\left(\frac{83}{308}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{135}{308}\right)$$ $$e\left(\frac{5}{77}\right)$$
$$\chi_{2001}(40,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{163}{308}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{117}{154}\right)$$ $$e\left(\frac{181}{308}\right)$$ $$e\left(\frac{151}{308}\right)$$ $$e\left(\frac{57}{308}\right)$$ $$e\left(\frac{81}{154}\right)$$ $$e\left(\frac{89}{308}\right)$$ $$e\left(\frac{9}{77}\right)$$
$$\chi_{2001}(43,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{283}{308}\right)$$ $$e\left(\frac{129}{154}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{233}{308}\right)$$ $$e\left(\frac{111}{308}\right)$$ $$e\left(\frac{201}{308}\right)$$ $$e\left(\frac{83}{154}\right)$$ $$e\left(\frac{249}{308}\right)$$ $$e\left(\frac{52}{77}\right)$$
$$\chi_{2001}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{223}{308}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{127}{154}\right)$$ $$e\left(\frac{53}{308}\right)$$ $$e\left(\frac{131}{308}\right)$$ $$e\left(\frac{129}{308}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{169}{308}\right)$$ $$e\left(\frac{69}{77}\right)$$
$$\chi_{2001}(76,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{308}\right)$$ $$e\left(\frac{37}{154}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{19}{154}\right)$$ $$e\left(\frac{111}{308}\right)$$ $$e\left(\frac{193}{308}\right)$$ $$e\left(\frac{183}{308}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{75}{308}\right)$$ $$e\left(\frac{37}{77}\right)$$
$$\chi_{2001}(79,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{271}{308}\right)$$ $$e\left(\frac{117}{154}\right)$$ $$e\left(\frac{38}{77}\right)$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{197}{308}\right)$$ $$e\left(\frac{115}{308}\right)$$ $$e\left(\frac{125}{308}\right)$$ $$e\left(\frac{129}{154}\right)$$ $$e\left(\frac{233}{308}\right)$$ $$e\left(\frac{40}{77}\right)$$
$$\chi_{2001}(97,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{281}{308}\right)$$ $$e\left(\frac{127}{154}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{111}{154}\right)$$ $$e\left(\frac{227}{308}\right)$$ $$e\left(\frac{9}{308}\right)$$ $$e\left(\frac{291}{308}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{195}{308}\right)$$ $$e\left(\frac{50}{77}\right)$$
$$\chi_{2001}(106,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{308}\right)$$ $$e\left(\frac{71}{154}\right)$$ $$e\left(\frac{2}{77}\right)$$ $$e\left(\frac{153}{154}\right)$$ $$e\left(\frac{213}{308}\right)$$ $$e\left(\frac{79}{308}\right)$$ $$e\left(\frac{193}{308}\right)$$ $$e\left(\frac{23}{154}\right)$$ $$e\left(\frac{69}{308}\right)$$ $$e\left(\frac{71}{77}\right)$$
$$\chi_{2001}(130,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{308}\right)$$ $$e\left(\frac{3}{154}\right)$$ $$e\left(\frac{76}{77}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{9}{308}\right)$$ $$e\left(\frac{307}{308}\right)$$ $$e\left(\frac{173}{308}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{81}{308}\right)$$ $$e\left(\frac{3}{77}\right)$$
$$\chi_{2001}(148,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{139}{308}\right)$$ $$e\left(\frac{139}{154}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{113}{154}\right)$$ $$e\left(\frac{109}{308}\right)$$ $$e\left(\frac{159}{308}\right)$$ $$e\left(\frac{213}{308}\right)$$ $$e\left(\frac{19}{154}\right)$$ $$e\left(\frac{57}{308}\right)$$ $$e\left(\frac{62}{77}\right)$$
$$\chi_{2001}(166,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{215}{308}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{31}{77}\right)$$ $$e\left(\frac{23}{154}\right)$$ $$e\left(\frac{29}{308}\right)$$ $$e\left(\frac{31}{308}\right)$$ $$e\left(\frac{181}{308}\right)$$ $$e\left(\frac{87}{154}\right)$$ $$e\left(\frac{261}{308}\right)$$ $$e\left(\frac{61}{77}\right)$$
$$\chi_{2001}(172,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{109}{308}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{15}{77}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{19}{308}\right)$$ $$e\left(\frac{169}{308}\right)$$ $$e\left(\frac{23}{308}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{171}{308}\right)$$ $$e\left(\frac{32}{77}\right)$$
$$\chi_{2001}(205,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{308}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{29}{77}\right)$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{201}{308}\right)$$ $$e\left(\frac{183}{308}\right)$$ $$e\left(\frac{65}{308}\right)$$ $$e\left(\frac{141}{154}\right)$$ $$e\left(\frac{269}{308}\right)$$ $$e\left(\frac{67}{77}\right)$$
$$\chi_{2001}(214,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{191}{308}\right)$$ $$e\left(\frac{37}{154}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{19}{154}\right)$$ $$e\left(\frac{265}{308}\right)$$ $$e\left(\frac{39}{308}\right)$$ $$e\left(\frac{29}{308}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{229}{308}\right)$$ $$e\left(\frac{37}{77}\right)$$
$$\chi_{2001}(217,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{227}{308}\right)$$ $$e\left(\frac{73}{154}\right)$$ $$e\left(\frac{27}{77}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{65}{308}\right)$$ $$e\left(\frac{27}{308}\right)$$ $$e\left(\frac{257}{308}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{73}{77}\right)$$
$$\chi_{2001}(235,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{308}\right)$$ $$e\left(\frac{83}{154}\right)$$ $$e\left(\frac{75}{77}\right)$$ $$e\left(\frac{1}{154}\right)$$ $$e\left(\frac{249}{308}\right)$$ $$e\left(\frac{75}{308}\right)$$ $$e\left(\frac{269}{308}\right)$$ $$e\left(\frac{131}{154}\right)$$ $$e\left(\frac{85}{308}\right)$$ $$e\left(\frac{6}{77}\right)$$
$$\chi_{2001}(247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{185}{308}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{95}{154}\right)$$ $$e\left(\frac{247}{308}\right)$$ $$e\left(\frac{41}{308}\right)$$ $$e\left(\frac{299}{308}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{67}{308}\right)$$ $$e\left(\frac{31}{77}\right)$$
$$\chi_{2001}(250,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{261}{308}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{167}{308}\right)$$ $$e\left(\frac{221}{308}\right)$$ $$e\left(\frac{267}{308}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{271}{308}\right)$$ $$e\left(\frac{30}{77}\right)$$
$$\chi_{2001}(304,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{171}{308}\right)$$ $$e\left(\frac{17}{154}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{205}{308}\right)$$ $$e\left(\frac{251}{308}\right)$$ $$e\left(\frac{5}{308}\right)$$ $$e\left(\frac{153}{154}\right)$$ $$e\left(\frac{305}{308}\right)$$ $$e\left(\frac{17}{77}\right)$$
$$\chi_{2001}(316,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{97}{308}\right)$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{19}{77}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{291}{308}\right)$$ $$e\left(\frac{173}{308}\right)$$ $$e\left(\frac{255}{308}\right)$$ $$e\left(\frac{103}{154}\right)$$ $$e\left(\frac{155}{308}\right)$$ $$e\left(\frac{20}{77}\right)$$
$$\chi_{2001}(337,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{289}{308}\right)$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{32}{77}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{251}{308}\right)$$ $$e\left(\frac{109}{308}\right)$$ $$e\left(\frac{239}{308}\right)$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{103}{308}\right)$$ $$e\left(\frac{58}{77}\right)$$
$$\chi_{2001}(379,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{263}{308}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{15}{77}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{173}{308}\right)$$ $$e\left(\frac{15}{308}\right)$$ $$e\left(\frac{177}{308}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{17}{308}\right)$$ $$e\left(\frac{32}{77}\right)$$
$$\chi_{2001}(385,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{229}{308}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{52}{77}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{71}{308}\right)$$ $$e\left(\frac{129}{308}\right)$$ $$e\left(\frac{167}{308}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{23}{308}\right)$$ $$e\left(\frac{75}{77}\right)$$
$$\chi_{2001}(388,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{107}{308}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{13}{308}\right)$$ $$e\left(\frac{67}{308}\right)$$ $$e\left(\frac{113}{308}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{117}{308}\right)$$ $$e\left(\frac{30}{77}\right)$$
$$\chi_{2001}(421,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{213}{308}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{23}{308}\right)$$ $$e\left(\frac{237}{308}\right)$$ $$e\left(\frac{271}{308}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{207}{308}\right)$$ $$e\left(\frac{59}{77}\right)$$
$$\chi_{2001}(424,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{205}{308}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{307}{308}\right)$$ $$e\left(\frac{137}{308}\right)$$ $$e\left(\frac{15}{308}\right)$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{299}{308}\right)$$ $$e\left(\frac{51}{77}\right)$$
$$\chi_{2001}(433,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{215}{308}\right)$$ $$e\left(\frac{113}{308}\right)$$ $$e\left(\frac{163}{308}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{87}{308}\right)$$ $$e\left(\frac{46}{77}\right)$$
$$\chi_{2001}(454,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{295}{308}\right)$$ $$e\left(\frac{141}{154}\right)$$ $$e\left(\frac{30}{77}\right)$$ $$e\left(\frac{139}{154}\right)$$ $$e\left(\frac{269}{308}\right)$$ $$e\left(\frac{107}{308}\right)$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{37}{154}\right)$$ $$e\left(\frac{265}{308}\right)$$ $$e\left(\frac{64}{77}\right)$$
$$\chi_{2001}(475,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{135}{308}\right)$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{32}{77}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{97}{308}\right)$$ $$e\left(\frac{263}{308}\right)$$ $$e\left(\frac{85}{308}\right)$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{257}{308}\right)$$ $$e\left(\frac{58}{77}\right)$$