Properties

 Label 75.l Modulus $75$ Conductor $75$ Order $20$ Real no Primitive yes Minimal yes Parity even

Related objects

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

H = DirichletGroup(75, base_ring=CyclotomicField(20))

M = H._module

chi = DirichletCharacter(H, M([10,1]))

chi.galois_orbit()

[g,chi] = znchar(Mod(2,75))

order = charorder(g,chi)

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

Basic properties

 Modulus: $$75$$ Conductor: $$75$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$20$$ 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: $$\Q(\zeta_{20})$$ Fixed field: $$\Q(\zeta_{75})^+$$

Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{75}(2,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{75}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{75}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{75}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-i$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{75}(38,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{75}(47,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{75}(53,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{75}(62,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$