# Properties

 Label 51.i Modulus $51$ Conductor $51$ Order $16$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(51, base_ring=CyclotomicField(16))

M = H._module

chi = DirichletCharacter(H, M([8,5]))

chi.galois_orbit()

[g,chi] = znchar(Mod(5,51))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$51$$ Conductor: $$51$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$16$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes 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: $$\Q(\zeta_{16})$$ Fixed field: $$\Q(\zeta_{51})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{51}(5,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{51}(11,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$-1$$
$$\chi_{51}(14,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{51}(20,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$
$$\chi_{51}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{51}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{51}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{51}(44,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$