# Properties

 Label 175.t Modulus $175$ Conductor $175$ Order $30$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(175, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,20]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(4,175))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$175$$ Conductor: $$175$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$30$$ 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: $$\Q(\zeta_{15})$$ Fixed field: 30.30.35434884492252294752034913472016341984272003173828125.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$16$$
$$\chi_{175}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{175}(9,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$
$$\chi_{175}(39,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{175}(44,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{175}(79,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{175}(109,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{15}\right)$$
$$\chi_{175}(114,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{175}(144,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$