Properties

Label 78385.in
Modulus $78385$
Conductor $78385$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78385, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,14,15]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(1012,78385))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(78385\)
Conductor: \(78385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{78385}(1012,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{78385}(5413,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{78385}(6698,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{78385}(11838,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{78385}(15147,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{78385}(21572,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{78385}(22118,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{78385}(36253,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{78385}(42678,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{78385}(43417,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{78385}(49842,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{78385}(62692,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{78385}(63977,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{78385}(64523,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{78385}(69117,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{78385}(70948,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{5}{12}\right)\)