# Properties

 Label 3549.dg Modulus $3549$ Conductor $1183$ Order $78$ 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(3549, base_ring=CyclotomicField(78))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,52,9]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(25,3549))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$3549$$ Conductor: $$1183$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$78$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1183.bs 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: $\Q(\zeta_{39})$ Fixed field: Number field defined by a degree 78 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$
$$\chi_{3549}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{3549}(142,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{26}\right)$$
$$\chi_{3549}(298,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{9}{26}\right)$$
$$\chi_{3549}(415,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{21}{26}\right)$$
$$\chi_{3549}(571,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{26}\right)$$
$$\chi_{3549}(688,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{3549}(961,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{25}{26}\right)$$
$$\chi_{3549}(1117,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{3549}(1234,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{26}\right)$$
$$\chi_{3549}(1390,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{26}\right)$$
$$\chi_{3549}(1507,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{3}{26}\right)$$
$$\chi_{3549}(1663,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{26}\right)$$
$$\chi_{3549}(1780,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{26}\right)$$
$$\chi_{3549}(1936,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{21}{26}\right)$$
$$\chi_{3549}(2053,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{3549}(2209,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{3549}(2326,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{9}{26}\right)$$
$$\chi_{3549}(2482,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{25}{26}\right)$$
$$\chi_{3549}(2599,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{26}\right)$$
$$\chi_{3549}(2755,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{26}\right)$$
$$\chi_{3549}(3028,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{26}\right)$$
$$\chi_{3549}(3145,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{3549}(3301,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{26}\right)$$
$$\chi_{3549}(3418,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{26}\right)$$