# Properties

 Label 8470.bi Modulus $8470$ Conductor $55$ Order $20$ Real no Primitive no Minimal no Parity even

# Learn more

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

sage: H = DirichletGroup(8470, base_ring=CyclotomicField(20))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([15,0,2]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(1443,8470))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$8470$$ Conductor: $$55$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$20$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 55.l sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{20})$$ Fixed field: $$\Q(\zeta_{55})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$9$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$ $$37$$
$$\chi_{8470}(1443,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$i$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{8470}(1667,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{8470}(1933,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$
$$\chi_{8470}(3137,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$
$$\chi_{8470}(3627,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{8470}(4033,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{8470}(5727,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{8470}(8443,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$