# Properties

 Label 297.v Modulus $297$ Conductor $297$ Order $90$ Real no Primitive yes Minimal yes Parity odd

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(90))

M = H._module

chi = DirichletCharacter(H, M([25,36]))

chi.galois_orbit()

[g,chi] = znchar(Mod(5,297))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$297$$ Conductor: $$297$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$90$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes 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: $\Q(\zeta_{45})$ Fixed field: Number field defined by a degree 90 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{297}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{297}(14,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{297}(20,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{297}(38,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{297}(47,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{297}(59,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{297}(86,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{297}(92,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{297}(104,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{297}(113,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{297}(119,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{297}(137,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{297}(146,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{297}(158,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{297}(185,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{297}(191,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{297}(203,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{297}(212,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{297}(218,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{297}(236,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{297}(245,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{297}(257,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{297}(284,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{297}(290,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$