# Properties

 Label 1728.bz Modulus $1728$ Conductor $576$ Order $48$ Real no Primitive no Minimal no Parity odd

# Related objects

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

sage: H = DirichletGroup(1728, base_ring=CyclotomicField(48))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,9,40]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(125,1728))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1728$$ Conductor: $$576$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$48$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 576.bn sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{48})$$ Fixed field: Number field defined by a degree 48 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{1728}(125,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(197,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(341,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(413,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(557,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(629,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(773,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(845,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(989,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(1061,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(1205,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(1277,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(1421,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(1493,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1728}(1637,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(1709,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$