# Properties

 Label 3549.cu Modulus $3549$ Conductor $1183$ Order $52$ 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(52))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,26,49]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(34,3549))

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: $$52$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1183.bn 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: $\Q(\zeta_{52})$ Fixed field: Number field defined by a degree 52 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$
$$\chi_{3549}(34,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$-i$$ $$e\left(\frac{45}{52}\right)$$
$$\chi_{3549}(265,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$i$$ $$e\left(\frac{31}{52}\right)$$
$$\chi_{3549}(307,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$-i$$ $$e\left(\frac{25}{52}\right)$$
$$\chi_{3549}(538,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$i$$ $$e\left(\frac{51}{52}\right)$$
$$\chi_{3549}(580,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$-i$$ $$e\left(\frac{5}{52}\right)$$
$$\chi_{3549}(811,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$i$$ $$e\left(\frac{19}{52}\right)$$
$$\chi_{3549}(853,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$-i$$ $$e\left(\frac{37}{52}\right)$$
$$\chi_{3549}(1126,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$-i$$ $$e\left(\frac{17}{52}\right)$$
$$\chi_{3549}(1357,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$i$$ $$e\left(\frac{7}{52}\right)$$
$$\chi_{3549}(1399,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$-i$$ $$e\left(\frac{49}{52}\right)$$
$$\chi_{3549}(1630,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$i$$ $$e\left(\frac{27}{52}\right)$$
$$\chi_{3549}(1672,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$-i$$ $$e\left(\frac{29}{52}\right)$$
$$\chi_{3549}(1903,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$i$$ $$e\left(\frac{47}{52}\right)$$
$$\chi_{3549}(1945,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$-i$$ $$e\left(\frac{9}{52}\right)$$
$$\chi_{3549}(2176,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$i$$ $$e\left(\frac{15}{52}\right)$$
$$\chi_{3549}(2218,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$-i$$ $$e\left(\frac{41}{52}\right)$$
$$\chi_{3549}(2449,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$i$$ $$e\left(\frac{35}{52}\right)$$
$$\chi_{3549}(2491,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$-i$$ $$e\left(\frac{21}{52}\right)$$
$$\chi_{3549}(2722,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$i$$ $$e\left(\frac{3}{52}\right)$$
$$\chi_{3549}(2764,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$-i$$ $$e\left(\frac{1}{52}\right)$$
$$\chi_{3549}(2995,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$i$$ $$e\left(\frac{23}{52}\right)$$
$$\chi_{3549}(3037,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$-i$$ $$e\left(\frac{33}{52}\right)$$
$$\chi_{3549}(3268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$i$$ $$e\left(\frac{43}{52}\right)$$
$$\chi_{3549}(3541,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$i$$ $$e\left(\frac{11}{52}\right)$$