Properties

Label 2304.cc
Modulus $2304$
Conductor $2304$
Order $192$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(192))
 
M = H._module
 
chi = DirichletCharacter(H, M([96,183,128]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(43,2304))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2304\)
Conductor: \(2304\)
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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{2304}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{55}{192}\right)\) \(e\left(\frac{67}{96}\right)\) \(e\left(\frac{35}{192}\right)\) \(e\left(\frac{25}{192}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{27}{64}\right)\) \(e\left(\frac{17}{96}\right)\) \(e\left(\frac{55}{96}\right)\) \(e\left(\frac{173}{192}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{2304}(67,\cdot)\) \(-1\) \(1\) \(e\left(\frac{89}{192}\right)\) \(e\left(\frac{77}{96}\right)\) \(e\left(\frac{109}{192}\right)\) \(e\left(\frac{23}{192}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{53}{64}\right)\) \(e\left(\frac{31}{96}\right)\) \(e\left(\frac{89}{96}\right)\) \(e\left(\frac{67}{192}\right)\) \(e\left(\frac{13}{24}\right)\)
\(\chi_{2304}(115,\cdot)\) \(-1\) \(1\) \(e\left(\frac{109}{192}\right)\) \(e\left(\frac{49}{96}\right)\) \(e\left(\frac{17}{192}\right)\) \(e\left(\frac{67}{192}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{57}{64}\right)\) \(e\left(\frac{11}{96}\right)\) \(e\left(\frac{13}{96}\right)\) \(e\left(\frac{95}{192}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{2304}(139,\cdot)\) \(-1\) \(1\) \(e\left(\frac{95}{192}\right)\) \(e\left(\frac{11}{96}\right)\) \(e\left(\frac{43}{192}\right)\) \(e\left(\frac{113}{192}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{35}{64}\right)\) \(e\left(\frac{73}{96}\right)\) \(e\left(\frac{95}{96}\right)\) \(e\left(\frac{37}{192}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{2304}(187,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{192}\right)\) \(e\left(\frac{79}{96}\right)\) \(e\left(\frac{47}{192}\right)\) \(e\left(\frac{61}{192}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{53}{96}\right)\) \(e\left(\frac{19}{96}\right)\) \(e\left(\frac{161}{192}\right)\) \(e\left(\frac{23}{24}\right)\)
\(\chi_{2304}(211,\cdot)\) \(-1\) \(1\) \(e\left(\frac{149}{192}\right)\) \(e\left(\frac{89}{96}\right)\) \(e\left(\frac{25}{192}\right)\) \(e\left(\frac{155}{192}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{64}\right)\) \(e\left(\frac{67}{96}\right)\) \(e\left(\frac{53}{96}\right)\) \(e\left(\frac{151}{192}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{2304}(259,\cdot)\) \(-1\) \(1\) \(e\left(\frac{169}{192}\right)\) \(e\left(\frac{61}{96}\right)\) \(e\left(\frac{125}{192}\right)\) \(e\left(\frac{7}{192}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{64}\right)\) \(e\left(\frac{47}{96}\right)\) \(e\left(\frac{73}{96}\right)\) \(e\left(\frac{179}{192}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{2304}(283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{192}\right)\) \(e\left(\frac{23}{96}\right)\) \(e\left(\frac{55}{192}\right)\) \(e\left(\frac{149}{192}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{15}{64}\right)\) \(e\left(\frac{13}{96}\right)\) \(e\left(\frac{59}{96}\right)\) \(e\left(\frac{25}{192}\right)\) \(e\left(\frac{7}{24}\right)\)
\(\chi_{2304}(331,\cdot)\) \(-1\) \(1\) \(e\left(\frac{175}{192}\right)\) \(e\left(\frac{91}{96}\right)\) \(e\left(\frac{59}{192}\right)\) \(e\left(\frac{97}{192}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{51}{64}\right)\) \(e\left(\frac{89}{96}\right)\) \(e\left(\frac{79}{96}\right)\) \(e\left(\frac{149}{192}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{2304}(355,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{192}\right)\) \(e\left(\frac{5}{96}\right)\) \(e\left(\frac{133}{192}\right)\) \(e\left(\frac{95}{192}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{13}{64}\right)\) \(e\left(\frac{7}{96}\right)\) \(e\left(\frac{17}{96}\right)\) \(e\left(\frac{43}{192}\right)\) \(e\left(\frac{13}{24}\right)\)
\(\chi_{2304}(403,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{192}\right)\) \(e\left(\frac{73}{96}\right)\) \(e\left(\frac{41}{192}\right)\) \(e\left(\frac{139}{192}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{17}{64}\right)\) \(e\left(\frac{83}{96}\right)\) \(e\left(\frac{37}{96}\right)\) \(e\left(\frac{71}{192}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{2304}(427,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{192}\right)\) \(e\left(\frac{35}{96}\right)\) \(e\left(\frac{67}{192}\right)\) \(e\left(\frac{185}{192}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{59}{64}\right)\) \(e\left(\frac{49}{96}\right)\) \(e\left(\frac{23}{96}\right)\) \(e\left(\frac{13}{192}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{2304}(475,\cdot)\) \(-1\) \(1\) \(e\left(\frac{139}{192}\right)\) \(e\left(\frac{7}{96}\right)\) \(e\left(\frac{71}{192}\right)\) \(e\left(\frac{133}{192}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{31}{64}\right)\) \(e\left(\frac{29}{96}\right)\) \(e\left(\frac{43}{96}\right)\) \(e\left(\frac{137}{192}\right)\) \(e\left(\frac{23}{24}\right)\)
\(\chi_{2304}(499,\cdot)\) \(-1\) \(1\) \(e\left(\frac{77}{192}\right)\) \(e\left(\frac{17}{96}\right)\) \(e\left(\frac{49}{192}\right)\) \(e\left(\frac{35}{192}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{25}{64}\right)\) \(e\left(\frac{43}{96}\right)\) \(e\left(\frac{77}{96}\right)\) \(e\left(\frac{127}{192}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{2304}(547,\cdot)\) \(-1\) \(1\) \(e\left(\frac{97}{192}\right)\) \(e\left(\frac{85}{96}\right)\) \(e\left(\frac{149}{192}\right)\) \(e\left(\frac{79}{192}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{29}{64}\right)\) \(e\left(\frac{23}{96}\right)\) \(e\left(\frac{1}{96}\right)\) \(e\left(\frac{155}{192}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{2304}(571,\cdot)\) \(-1\) \(1\) \(e\left(\frac{179}{192}\right)\) \(e\left(\frac{47}{96}\right)\) \(e\left(\frac{79}{192}\right)\) \(e\left(\frac{29}{192}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{39}{64}\right)\) \(e\left(\frac{85}{96}\right)\) \(e\left(\frac{83}{96}\right)\) \(e\left(\frac{1}{192}\right)\) \(e\left(\frac{7}{24}\right)\)
\(\chi_{2304}(619,\cdot)\) \(-1\) \(1\) \(e\left(\frac{103}{192}\right)\) \(e\left(\frac{19}{96}\right)\) \(e\left(\frac{83}{192}\right)\) \(e\left(\frac{169}{192}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{11}{64}\right)\) \(e\left(\frac{65}{96}\right)\) \(e\left(\frac{7}{96}\right)\) \(e\left(\frac{125}{192}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{2304}(643,\cdot)\) \(-1\) \(1\) \(e\left(\frac{137}{192}\right)\) \(e\left(\frac{29}{96}\right)\) \(e\left(\frac{157}{192}\right)\) \(e\left(\frac{167}{192}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{79}{96}\right)\) \(e\left(\frac{41}{96}\right)\) \(e\left(\frac{19}{192}\right)\) \(e\left(\frac{13}{24}\right)\)
\(\chi_{2304}(691,\cdot)\) \(-1\) \(1\) \(e\left(\frac{157}{192}\right)\) \(e\left(\frac{1}{96}\right)\) \(e\left(\frac{65}{192}\right)\) \(e\left(\frac{19}{192}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{41}{64}\right)\) \(e\left(\frac{59}{96}\right)\) \(e\left(\frac{61}{96}\right)\) \(e\left(\frac{47}{192}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{2304}(715,\cdot)\) \(-1\) \(1\) \(e\left(\frac{143}{192}\right)\) \(e\left(\frac{59}{96}\right)\) \(e\left(\frac{91}{192}\right)\) \(e\left(\frac{65}{192}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{19}{64}\right)\) \(e\left(\frac{25}{96}\right)\) \(e\left(\frac{47}{96}\right)\) \(e\left(\frac{181}{192}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{2304}(763,\cdot)\) \(-1\) \(1\) \(e\left(\frac{67}{192}\right)\) \(e\left(\frac{31}{96}\right)\) \(e\left(\frac{95}{192}\right)\) \(e\left(\frac{13}{192}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{55}{64}\right)\) \(e\left(\frac{5}{96}\right)\) \(e\left(\frac{67}{96}\right)\) \(e\left(\frac{113}{192}\right)\) \(e\left(\frac{23}{24}\right)\)
\(\chi_{2304}(787,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{192}\right)\) \(e\left(\frac{41}{96}\right)\) \(e\left(\frac{73}{192}\right)\) \(e\left(\frac{107}{192}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{49}{64}\right)\) \(e\left(\frac{19}{96}\right)\) \(e\left(\frac{5}{96}\right)\) \(e\left(\frac{103}{192}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{2304}(835,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{192}\right)\) \(e\left(\frac{13}{96}\right)\) \(e\left(\frac{173}{192}\right)\) \(e\left(\frac{151}{192}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{53}{64}\right)\) \(e\left(\frac{95}{96}\right)\) \(e\left(\frac{25}{96}\right)\) \(e\left(\frac{131}{192}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{2304}(859,\cdot)\) \(-1\) \(1\) \(e\left(\frac{107}{192}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{103}{192}\right)\) \(e\left(\frac{101}{192}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{63}{64}\right)\) \(e\left(\frac{61}{96}\right)\) \(e\left(\frac{11}{96}\right)\) \(e\left(\frac{169}{192}\right)\) \(e\left(\frac{7}{24}\right)\)
\(\chi_{2304}(907,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{192}\right)\) \(e\left(\frac{43}{96}\right)\) \(e\left(\frac{107}{192}\right)\) \(e\left(\frac{49}{192}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{35}{64}\right)\) \(e\left(\frac{41}{96}\right)\) \(e\left(\frac{31}{96}\right)\) \(e\left(\frac{101}{192}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{2304}(931,\cdot)\) \(-1\) \(1\) \(e\left(\frac{65}{192}\right)\) \(e\left(\frac{53}{96}\right)\) \(e\left(\frac{181}{192}\right)\) \(e\left(\frac{47}{192}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{61}{64}\right)\) \(e\left(\frac{55}{96}\right)\) \(e\left(\frac{65}{96}\right)\) \(e\left(\frac{187}{192}\right)\) \(e\left(\frac{13}{24}\right)\)
\(\chi_{2304}(979,\cdot)\) \(-1\) \(1\) \(e\left(\frac{85}{192}\right)\) \(e\left(\frac{25}{96}\right)\) \(e\left(\frac{89}{192}\right)\) \(e\left(\frac{91}{192}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{64}\right)\) \(e\left(\frac{35}{96}\right)\) \(e\left(\frac{85}{96}\right)\) \(e\left(\frac{23}{192}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{2304}(1003,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{192}\right)\) \(e\left(\frac{83}{96}\right)\) \(e\left(\frac{115}{192}\right)\) \(e\left(\frac{137}{192}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{43}{64}\right)\) \(e\left(\frac{1}{96}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{157}{192}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{2304}(1051,\cdot)\) \(-1\) \(1\) \(e\left(\frac{187}{192}\right)\) \(e\left(\frac{55}{96}\right)\) \(e\left(\frac{119}{192}\right)\) \(e\left(\frac{85}{192}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{15}{64}\right)\) \(e\left(\frac{77}{96}\right)\) \(e\left(\frac{91}{96}\right)\) \(e\left(\frac{89}{192}\right)\) \(e\left(\frac{23}{24}\right)\)
\(\chi_{2304}(1075,\cdot)\) \(-1\) \(1\) \(e\left(\frac{125}{192}\right)\) \(e\left(\frac{65}{96}\right)\) \(e\left(\frac{97}{192}\right)\) \(e\left(\frac{179}{192}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{9}{64}\right)\) \(e\left(\frac{91}{96}\right)\) \(e\left(\frac{29}{96}\right)\) \(e\left(\frac{79}{192}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{2304}(1123,\cdot)\) \(-1\) \(1\) \(e\left(\frac{145}{192}\right)\) \(e\left(\frac{37}{96}\right)\) \(e\left(\frac{5}{192}\right)\) \(e\left(\frac{31}{192}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{13}{64}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{49}{96}\right)\) \(e\left(\frac{107}{192}\right)\) \(e\left(\frac{5}{24}\right)\)