# Properties

 Label 5185.jj Modulus $5185$ Conductor $1037$ Order $120$ Real no Primitive no Minimal yes Parity odd

# Learn more about

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

sage: H = DirichletGroup(5185, base_ring=CyclotomicField(120))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,15,82]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(26,5185))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$5185$$ Conductor: $$1037$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$120$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1037.cm sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{120})$ Fixed field: Number field defined by a degree 120 polynomial (not computed)

## First 31 of 32 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{5185}(26,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(246,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(331,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(376,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(1141,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1226,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1386,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(1396,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1471,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1481,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(1596,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(1641,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1691,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(1726,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1776,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(1946,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(2031,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(2361,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(2446,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(2491,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(2616,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(2701,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(2796,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(2841,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(2926,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(3691,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(4061,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(4146,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{5185}(4361,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(4666,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{5185}(4911,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$