Properties

Label 1600.cq
Modulus $1600$
Conductor $1600$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1600, base_ring=CyclotomicField(80))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,8]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(29,1600))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1600\)
Conductor: \(1600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

First 31 of 32 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{1600}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{3}{80}\right)\) \(e\left(\frac{17}{80}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{49}{80}\right)\) \(e\left(\frac{11}{80}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{23}{80}\right)\)
\(\chi_{1600}(69,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{57}{80}\right)\) \(e\left(\frac{3}{80}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{51}{80}\right)\) \(e\left(\frac{49}{80}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{37}{80}\right)\)
\(\chi_{1600}(109,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{31}{80}\right)\) \(e\left(\frac{69}{80}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{53}{80}\right)\) \(e\left(\frac{7}{80}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{51}{80}\right)\)
\(\chi_{1600}(189,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{59}{80}\right)\) \(e\left(\frac{41}{80}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{57}{80}\right)\) \(e\left(\frac{3}{80}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{79}{80}\right)\)
\(\chi_{1600}(229,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{33}{80}\right)\) \(e\left(\frac{27}{80}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{59}{80}\right)\) \(e\left(\frac{41}{80}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{13}{80}\right)\)
\(\chi_{1600}(269,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{7}{80}\right)\) \(e\left(\frac{13}{80}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{61}{80}\right)\) \(e\left(\frac{79}{80}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{27}{80}\right)\)
\(\chi_{1600}(309,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{61}{80}\right)\) \(e\left(\frac{79}{80}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{63}{80}\right)\) \(e\left(\frac{37}{80}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{41}{80}\right)\)
\(\chi_{1600}(389,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{9}{80}\right)\) \(e\left(\frac{51}{80}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{67}{80}\right)\) \(e\left(\frac{33}{80}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{69}{80}\right)\)
\(\chi_{1600}(429,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{63}{80}\right)\) \(e\left(\frac{37}{80}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{69}{80}\right)\) \(e\left(\frac{71}{80}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{3}{80}\right)\)
\(\chi_{1600}(469,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{37}{80}\right)\) \(e\left(\frac{23}{80}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{71}{80}\right)\) \(e\left(\frac{29}{80}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{17}{80}\right)\)
\(\chi_{1600}(509,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{11}{80}\right)\) \(e\left(\frac{9}{80}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{73}{80}\right)\) \(e\left(\frac{67}{80}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{31}{80}\right)\)
\(\chi_{1600}(589,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{39}{80}\right)\) \(e\left(\frac{61}{80}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{77}{80}\right)\) \(e\left(\frac{63}{80}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{59}{80}\right)\)
\(\chi_{1600}(629,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{13}{80}\right)\) \(e\left(\frac{47}{80}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{79}{80}\right)\) \(e\left(\frac{21}{80}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{73}{80}\right)\)
\(\chi_{1600}(669,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{67}{80}\right)\) \(e\left(\frac{33}{80}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{80}\right)\) \(e\left(\frac{59}{80}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{7}{80}\right)\)
\(\chi_{1600}(709,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{41}{80}\right)\) \(e\left(\frac{19}{80}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{80}\right)\) \(e\left(\frac{17}{80}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{21}{80}\right)\)
\(\chi_{1600}(789,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{69}{80}\right)\) \(e\left(\frac{71}{80}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{80}\right)\) \(e\left(\frac{13}{80}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{49}{80}\right)\)
\(\chi_{1600}(829,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{43}{80}\right)\) \(e\left(\frac{57}{80}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{80}\right)\) \(e\left(\frac{51}{80}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{63}{80}\right)\)
\(\chi_{1600}(869,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{17}{80}\right)\) \(e\left(\frac{43}{80}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{11}{80}\right)\) \(e\left(\frac{9}{80}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{77}{80}\right)\)
\(\chi_{1600}(909,\cdot)\) \(1\) \(1\) \(e\left(\frac{57}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{71}{80}\right)\) \(e\left(\frac{29}{80}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{13}{80}\right)\) \(e\left(\frac{47}{80}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{11}{80}\right)\)
\(\chi_{1600}(989,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{19}{80}\right)\) \(e\left(\frac{1}{80}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{17}{80}\right)\) \(e\left(\frac{43}{80}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{39}{80}\right)\)
\(\chi_{1600}(1029,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{73}{80}\right)\) \(e\left(\frac{67}{80}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{80}\right)\) \(e\left(\frac{1}{80}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{53}{80}\right)\)
\(\chi_{1600}(1069,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{47}{80}\right)\) \(e\left(\frac{53}{80}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{21}{80}\right)\) \(e\left(\frac{39}{80}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{67}{80}\right)\)
\(\chi_{1600}(1109,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{21}{80}\right)\) \(e\left(\frac{39}{80}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{23}{80}\right)\) \(e\left(\frac{77}{80}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{1}{80}\right)\)
\(\chi_{1600}(1189,\cdot)\) \(1\) \(1\) \(e\left(\frac{63}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{49}{80}\right)\) \(e\left(\frac{11}{80}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{27}{80}\right)\) \(e\left(\frac{73}{80}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{29}{80}\right)\)
\(\chi_{1600}(1229,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{23}{80}\right)\) \(e\left(\frac{77}{80}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{29}{80}\right)\) \(e\left(\frac{31}{80}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{43}{80}\right)\)
\(\chi_{1600}(1269,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{77}{80}\right)\) \(e\left(\frac{63}{80}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{31}{80}\right)\) \(e\left(\frac{69}{80}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{57}{80}\right)\)
\(\chi_{1600}(1309,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{51}{80}\right)\) \(e\left(\frac{49}{80}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{33}{80}\right)\) \(e\left(\frac{27}{80}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{71}{80}\right)\)
\(\chi_{1600}(1389,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{80}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{79}{80}\right)\) \(e\left(\frac{21}{80}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{37}{80}\right)\) \(e\left(\frac{23}{80}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{19}{80}\right)\)
\(\chi_{1600}(1429,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{80}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{53}{80}\right)\) \(e\left(\frac{7}{80}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{39}{80}\right)\) \(e\left(\frac{61}{80}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{33}{80}\right)\)
\(\chi_{1600}(1469,\cdot)\) \(1\) \(1\) \(e\left(\frac{69}{80}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{27}{80}\right)\) \(e\left(\frac{73}{80}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{41}{80}\right)\) \(e\left(\frac{19}{80}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{47}{80}\right)\)
\(\chi_{1600}(1509,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{80}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{1}{80}\right)\) \(e\left(\frac{59}{80}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{43}{80}\right)\) \(e\left(\frac{57}{80}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{61}{80}\right)\)