# Properties

 Label 1792.cd Modulus $1792$ Conductor $1792$ Order $192$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1792, base_ring=CyclotomicField(192))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([96,63,128]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(11,1792))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1792$$ Conductor: $$1792$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$192$$ 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: $\Q(\zeta_{192})$ Fixed field: Number field defined by a degree 192 polynomial (not computed)

## First 31 of 64 characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{1792}(11,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{125}{192}\right)$$ $$e\left(\frac{127}{192}\right)$$ $$e\left(\frac{29}{96}\right)$$ $$e\left(\frac{11}{192}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{73}{192}\right)$$ $$e\left(\frac{41}{96}\right)$$ $$e\left(\frac{31}{96}\right)$$
$$\chi_{1792}(51,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{55}{192}\right)$$ $$e\left(\frac{125}{192}\right)$$ $$e\left(\frac{55}{96}\right)$$ $$e\left(\frac{97}{192}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{155}{192}\right)$$ $$e\left(\frac{91}{96}\right)$$ $$e\left(\frac{29}{96}\right)$$
$$\chi_{1792}(67,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{192}\right)$$ $$e\left(\frac{25}{192}\right)$$ $$e\left(\frac{11}{96}\right)$$ $$e\left(\frac{173}{192}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{31}{192}\right)$$ $$e\left(\frac{95}{96}\right)$$ $$e\left(\frac{25}{96}\right)$$
$$\chi_{1792}(107,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{181}{192}\right)$$ $$e\left(\frac{167}{192}\right)$$ $$e\left(\frac{85}{96}\right)$$ $$e\left(\frac{19}{192}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{161}{192}\right)$$ $$e\left(\frac{1}{96}\right)$$ $$e\left(\frac{71}{96}\right)$$
$$\chi_{1792}(123,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{192}\right)$$ $$e\left(\frac{163}{192}\right)$$ $$e\left(\frac{41}{96}\right)$$ $$e\left(\frac{191}{192}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{133}{192}\right)$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{67}{96}\right)$$
$$\chi_{1792}(163,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{192}\right)$$ $$e\left(\frac{65}{192}\right)$$ $$e\left(\frac{67}{96}\right)$$ $$e\left(\frac{181}{192}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{119}{192}\right)$$ $$e\left(\frac{55}{96}\right)$$ $$e\left(\frac{65}{96}\right)$$
$$\chi_{1792}(179,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{192}\right)$$ $$e\left(\frac{157}{192}\right)$$ $$e\left(\frac{23}{96}\right)$$ $$e\left(\frac{65}{192}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{187}{192}\right)$$ $$e\left(\frac{59}{96}\right)$$ $$e\left(\frac{61}{96}\right)$$
$$\chi_{1792}(219,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{97}{192}\right)$$ $$e\left(\frac{11}{192}\right)$$ $$e\left(\frac{1}{96}\right)$$ $$e\left(\frac{7}{192}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{29}{192}\right)$$ $$e\left(\frac{61}{96}\right)$$ $$e\left(\frac{11}{96}\right)$$
$$\chi_{1792}(235,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{149}{192}\right)$$ $$e\left(\frac{7}{192}\right)$$ $$e\left(\frac{53}{96}\right)$$ $$e\left(\frac{179}{192}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{1}{192}\right)$$ $$e\left(\frac{65}{96}\right)$$ $$e\left(\frac{7}{96}\right)$$
$$\chi_{1792}(275,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{79}{192}\right)$$ $$e\left(\frac{5}{192}\right)$$ $$e\left(\frac{79}{96}\right)$$ $$e\left(\frac{73}{192}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{83}{192}\right)$$ $$e\left(\frac{19}{96}\right)$$ $$e\left(\frac{5}{96}\right)$$
$$\chi_{1792}(291,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{192}\right)$$ $$e\left(\frac{97}{192}\right)$$ $$e\left(\frac{35}{96}\right)$$ $$e\left(\frac{149}{192}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{151}{192}\right)$$ $$e\left(\frac{23}{96}\right)$$ $$e\left(\frac{1}{96}\right)$$
$$\chi_{1792}(331,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{192}\right)$$ $$e\left(\frac{47}{192}\right)$$ $$e\left(\frac{13}{96}\right)$$ $$e\left(\frac{187}{192}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{89}{192}\right)$$ $$e\left(\frac{25}{96}\right)$$ $$e\left(\frac{47}{96}\right)$$
$$\chi_{1792}(347,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{65}{192}\right)$$ $$e\left(\frac{43}{192}\right)$$ $$e\left(\frac{65}{96}\right)$$ $$e\left(\frac{167}{192}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{61}{192}\right)$$ $$e\left(\frac{29}{96}\right)$$ $$e\left(\frac{43}{96}\right)$$
$$\chi_{1792}(387,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{91}{192}\right)$$ $$e\left(\frac{137}{192}\right)$$ $$e\left(\frac{91}{96}\right)$$ $$e\left(\frac{157}{192}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{47}{192}\right)$$ $$e\left(\frac{79}{96}\right)$$ $$e\left(\frac{41}{96}\right)$$
$$\chi_{1792}(403,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{192}\right)$$ $$e\left(\frac{37}{192}\right)$$ $$e\left(\frac{47}{96}\right)$$ $$e\left(\frac{41}{192}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{115}{192}\right)$$ $$e\left(\frac{83}{96}\right)$$ $$e\left(\frac{37}{96}\right)$$
$$\chi_{1792}(443,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{121}{192}\right)$$ $$e\left(\frac{83}{192}\right)$$ $$e\left(\frac{25}{96}\right)$$ $$e\left(\frac{175}{192}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{149}{192}\right)$$ $$e\left(\frac{85}{96}\right)$$ $$e\left(\frac{83}{96}\right)$$
$$\chi_{1792}(459,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{173}{192}\right)$$ $$e\left(\frac{79}{192}\right)$$ $$e\left(\frac{77}{96}\right)$$ $$e\left(\frac{155}{192}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{121}{192}\right)$$ $$e\left(\frac{89}{96}\right)$$ $$e\left(\frac{79}{96}\right)$$
$$\chi_{1792}(499,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{103}{192}\right)$$ $$e\left(\frac{77}{192}\right)$$ $$e\left(\frac{7}{96}\right)$$ $$e\left(\frac{49}{192}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{11}{192}\right)$$ $$e\left(\frac{43}{96}\right)$$ $$e\left(\frac{77}{96}\right)$$
$$\chi_{1792}(515,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{59}{192}\right)$$ $$e\left(\frac{169}{192}\right)$$ $$e\left(\frac{59}{96}\right)$$ $$e\left(\frac{125}{192}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{79}{192}\right)$$ $$e\left(\frac{47}{96}\right)$$ $$e\left(\frac{73}{96}\right)$$
$$\chi_{1792}(555,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{192}\right)$$ $$e\left(\frac{119}{192}\right)$$ $$e\left(\frac{37}{96}\right)$$ $$e\left(\frac{163}{192}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{17}{192}\right)$$ $$e\left(\frac{49}{96}\right)$$ $$e\left(\frac{23}{96}\right)$$
$$\chi_{1792}(571,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{89}{192}\right)$$ $$e\left(\frac{115}{192}\right)$$ $$e\left(\frac{89}{96}\right)$$ $$e\left(\frac{143}{192}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{181}{192}\right)$$ $$e\left(\frac{53}{96}\right)$$ $$e\left(\frac{19}{96}\right)$$
$$\chi_{1792}(611,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{115}{192}\right)$$ $$e\left(\frac{17}{192}\right)$$ $$e\left(\frac{19}{96}\right)$$ $$e\left(\frac{133}{192}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{167}{192}\right)$$ $$e\left(\frac{7}{96}\right)$$ $$e\left(\frac{17}{96}\right)$$
$$\chi_{1792}(627,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{71}{192}\right)$$ $$e\left(\frac{109}{192}\right)$$ $$e\left(\frac{71}{96}\right)$$ $$e\left(\frac{17}{192}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{43}{192}\right)$$ $$e\left(\frac{11}{96}\right)$$ $$e\left(\frac{13}{96}\right)$$
$$\chi_{1792}(667,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{145}{192}\right)$$ $$e\left(\frac{155}{192}\right)$$ $$e\left(\frac{49}{96}\right)$$ $$e\left(\frac{151}{192}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{77}{192}\right)$$ $$e\left(\frac{13}{96}\right)$$ $$e\left(\frac{59}{96}\right)$$
$$\chi_{1792}(683,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{192}\right)$$ $$e\left(\frac{151}{192}\right)$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{131}{192}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{49}{192}\right)$$ $$e\left(\frac{17}{96}\right)$$ $$e\left(\frac{55}{96}\right)$$
$$\chi_{1792}(723,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{127}{192}\right)$$ $$e\left(\frac{149}{192}\right)$$ $$e\left(\frac{31}{96}\right)$$ $$e\left(\frac{25}{192}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{131}{192}\right)$$ $$e\left(\frac{67}{96}\right)$$ $$e\left(\frac{53}{96}\right)$$
$$\chi_{1792}(739,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{83}{192}\right)$$ $$e\left(\frac{49}{192}\right)$$ $$e\left(\frac{83}{96}\right)$$ $$e\left(\frac{101}{192}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{7}{192}\right)$$ $$e\left(\frac{71}{96}\right)$$ $$e\left(\frac{49}{96}\right)$$
$$\chi_{1792}(779,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{61}{192}\right)$$ $$e\left(\frac{191}{192}\right)$$ $$e\left(\frac{61}{96}\right)$$ $$e\left(\frac{139}{192}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{137}{192}\right)$$ $$e\left(\frac{73}{96}\right)$$ $$e\left(\frac{95}{96}\right)$$
$$\chi_{1792}(795,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{113}{192}\right)$$ $$e\left(\frac{187}{192}\right)$$ $$e\left(\frac{17}{96}\right)$$ $$e\left(\frac{119}{192}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{109}{192}\right)$$ $$e\left(\frac{77}{96}\right)$$ $$e\left(\frac{91}{96}\right)$$
$$\chi_{1792}(835,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{139}{192}\right)$$ $$e\left(\frac{89}{192}\right)$$ $$e\left(\frac{43}{96}\right)$$ $$e\left(\frac{109}{192}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{95}{192}\right)$$ $$e\left(\frac{31}{96}\right)$$ $$e\left(\frac{89}{96}\right)$$
$$\chi_{1792}(851,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{95}{192}\right)$$ $$e\left(\frac{181}{192}\right)$$ $$e\left(\frac{95}{96}\right)$$ $$e\left(\frac{185}{192}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{163}{192}\right)$$ $$e\left(\frac{35}{96}\right)$$ $$e\left(\frac{85}{96}\right)$$