# Properties

 Label 384.x Modulus $384$ Conductor $384$ Order $32$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(384, base_ring=CyclotomicField(32))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1,16]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(5,384))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$384$$ Conductor: $$384$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$32$$ 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: $$\Q(\zeta_{32})$$ Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{384}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$i$$
$$\chi_{384}(29,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$-i$$
$$\chi_{384}(53,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$i$$
$$\chi_{384}(77,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$-i$$
$$\chi_{384}(101,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$i$$
$$\chi_{384}(125,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$-i$$
$$\chi_{384}(149,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$i$$
$$\chi_{384}(173,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$-i$$
$$\chi_{384}(197,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$i$$
$$\chi_{384}(221,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$-i$$
$$\chi_{384}(245,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$i$$
$$\chi_{384}(269,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$-i$$
$$\chi_{384}(293,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$i$$
$$\chi_{384}(317,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$-i$$
$$\chi_{384}(341,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$i$$
$$\chi_{384}(365,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$-i$$