Properties

Label 3600.fp
Modulus $3600$
Conductor $3600$
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(3600, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,45,10,9]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(83,3600))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3600\)
Conductor: \(3600\)
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 \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{3600}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{3600}(347,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{3600}(563,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{3600}(587,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{3600}(803,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{3600}(1067,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{3600}(1283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{3600}(1523,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{3600}(1787,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{3600}(2003,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{3600}(2027,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{3600}(2723,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{3600}(2747,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{3600}(2963,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{3600}(3227,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{3600}(3467,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{14}{15}\right)\)