# Properties

 Label 49.h Modulus $49$ Conductor $49$ Order $42$ Real no Primitive yes Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(49, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(3,49))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$49$$ Conductor: $$49$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ 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: $$\Q(\zeta_{21})$$ Fixed field: $$\Q(\zeta_{49})$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{49}(3,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$
$$\chi_{49}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$
$$\chi_{49}(10,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{49}(12,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{49}(17,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{49}(24,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{49}(26,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$
$$\chi_{49}(33,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$
$$\chi_{49}(38,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$
$$\chi_{49}(40,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$
$$\chi_{49}(45,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$
$$\chi_{49}(47,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$