# Properties

 Label 4017.dp Modulus $4017$ Conductor $4017$ Order $102$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(4017, base_ring=CyclotomicField(102))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([51,51,65]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(77,4017))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$4017$$ Conductor: $$4017$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$102$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes 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: $\Q(\zeta_{51})$ Fixed field: Number field defined by a degree 102 polynomial (not computed)

## First 31 of 32 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$14$$ $$16$$ $$17$$
$$\chi_{4017}(77,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{65}{102}\right)$$ $$e\left(\frac{5}{102}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{89}{102}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{11}{102}\right)$$
$$\chi_{4017}(623,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{1}{102}\right)$$ $$e\left(\frac{55}{102}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{61}{102}\right)$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{19}{102}\right)$$
$$\chi_{4017}(662,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{47}{102}\right)$$ $$e\left(\frac{35}{102}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{11}{102}\right)$$ $$e\left(\frac{21}{34}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{77}{102}\right)$$
$$\chi_{4017}(1013,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{19}{102}\right)$$ $$e\left(\frac{25}{102}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{37}{102}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{55}{102}\right)$$
$$\chi_{4017}(1208,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{41}{102}\right)$$ $$e\left(\frac{11}{102}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{53}{102}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{65}{102}\right)$$
$$\chi_{4017}(1247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{61}{102}\right)$$ $$e\left(\frac{91}{102}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{49}{102}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{37}{102}\right)$$
$$\chi_{4017}(1520,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{53}{102}\right)$$ $$e\left(\frac{59}{102}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{71}{102}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{89}{102}\right)$$
$$\chi_{4017}(1598,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{97}{102}\right)$$ $$e\left(\frac{31}{102}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{1}{102}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{7}{102}\right)$$
$$\chi_{4017}(1715,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{13}{102}\right)$$ $$e\left(\frac{1}{102}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{79}{102}\right)$$ $$e\left(\frac{21}{34}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{43}{102}\right)$$
$$\chi_{4017}(2027,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{49}{102}\right)$$ $$e\left(\frac{43}{102}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{21}{34}\right)$$ $$e\left(\frac{31}{102}\right)$$ $$e\left(\frac{19}{34}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{13}{102}\right)$$
$$\chi_{4017}(2066,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{83}{102}\right)$$ $$e\left(\frac{77}{102}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{21}{34}\right)$$ $$e\left(\frac{65}{102}\right)$$ $$e\left(\frac{19}{34}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{47}{102}\right)$$
$$\chi_{4017}(2105,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{79}{102}\right)$$ $$e\left(\frac{61}{102}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{25}{102}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{73}{102}\right)$$
$$\chi_{4017}(2144,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{29}{102}\right)$$ $$e\left(\frac{65}{102}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{35}{102}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{41}{102}\right)$$
$$\chi_{4017}(2183,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{89}{102}\right)$$ $$e\left(\frac{101}{102}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{9}{34}\right)$$ $$e\left(\frac{23}{102}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{59}{102}\right)$$
$$\chi_{4017}(2417,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{11}{102}\right)$$ $$e\left(\frac{95}{102}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{59}{102}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{5}{102}\right)$$
$$\chi_{4017}(2456,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{23}{102}\right)$$ $$e\left(\frac{41}{102}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{77}{102}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{29}{102}\right)$$
$$\chi_{4017}(2534,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{101}{102}\right)$$ $$e\left(\frac{47}{102}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{19}{34}\right)$$ $$e\left(\frac{41}{102}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{83}{102}\right)$$
$$\chi_{4017}(2573,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{95}{102}\right)$$ $$e\left(\frac{23}{102}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{83}{102}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{71}{102}\right)$$
$$\chi_{4017}(2690,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{25}{102}\right)$$ $$e\left(\frac{49}{102}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{97}{102}\right)$$ $$e\left(\frac{9}{34}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{67}{102}\right)$$
$$\chi_{4017}(2729,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{7}{102}\right)$$ $$e\left(\frac{79}{102}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{19}{102}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{31}{102}\right)$$
$$\chi_{4017}(2846,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{73}{102}\right)$$ $$e\left(\frac{37}{102}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{67}{102}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{61}{102}\right)$$
$$\chi_{4017}(2924,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{31}{102}\right)$$ $$e\left(\frac{73}{102}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{55}{102}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{79}{102}\right)$$
$$\chi_{4017}(3041,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{59}{102}\right)$$ $$e\left(\frac{83}{102}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{29}{102}\right)$$ $$e\left(\frac{9}{34}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{101}{102}\right)$$
$$\chi_{4017}(3236,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{77}{102}\right)$$ $$e\left(\frac{53}{102}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{5}{102}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{35}{102}\right)$$
$$\chi_{4017}(3392,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{55}{102}\right)$$ $$e\left(\frac{67}{102}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{9}{34}\right)$$ $$e\left(\frac{91}{102}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{25}{102}\right)$$
$$\chi_{4017}(3470,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{67}{102}\right)$$ $$e\left(\frac{13}{102}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{19}{34}\right)$$ $$e\left(\frac{7}{102}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{49}{102}\right)$$
$$\chi_{4017}(3587,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{71}{102}\right)$$ $$e\left(\frac{29}{102}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{47}{102}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{23}{102}\right)$$
$$\chi_{4017}(3626,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{43}{102}\right)$$ $$e\left(\frac{19}{102}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{73}{102}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{1}{102}\right)$$
$$\chi_{4017}(3704,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{37}{102}\right)$$ $$e\left(\frac{97}{102}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{13}{102}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{91}{102}\right)$$
$$\chi_{4017}(3743,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{5}{102}\right)$$ $$e\left(\frac{71}{102}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{101}{102}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{95}{102}\right)$$
$$\chi_{4017}(3782,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{35}{102}\right)$$ $$e\left(\frac{89}{102}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{95}{102}\right)$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{53}{102}\right)$$