Properties

Label 3600.227
Modulus $3600$
Conductor $3600$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,45,10,3]))
 
pari: [g,chi] = znchar(Mod(227,3600))
 

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)
 

Galois orbit 3600.ft

\(\chi_{3600}(203,\cdot)\) \(\chi_{3600}(227,\cdot)\) \(\chi_{3600}(923,\cdot)\) \(\chi_{3600}(947,\cdot)\) \(\chi_{3600}(1163,\cdot)\) \(\chi_{3600}(1427,\cdot)\) \(\chi_{3600}(1667,\cdot)\) \(\chi_{3600}(1883,\cdot)\) \(\chi_{3600}(2147,\cdot)\) \(\chi_{3600}(2363,\cdot)\) \(\chi_{3600}(2387,\cdot)\) \(\chi_{3600}(2603,\cdot)\) \(\chi_{3600}(2867,\cdot)\) \(\chi_{3600}(3083,\cdot)\) \(\chi_{3600}(3323,\cdot)\) \(\chi_{3600}(3587,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3151,901,2801,577)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(227, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(227,a) \;\) at \(\;a = \) e.g. 2