Properties

Label 3600.83
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]))
 
pari: [g,chi] = znchar(Mod(83,3600))
 

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.fp

\(\chi_{3600}(83,\cdot)\) \(\chi_{3600}(347,\cdot)\) \(\chi_{3600}(563,\cdot)\) \(\chi_{3600}(587,\cdot)\) \(\chi_{3600}(803,\cdot)\) \(\chi_{3600}(1067,\cdot)\) \(\chi_{3600}(1283,\cdot)\) \(\chi_{3600}(1523,\cdot)\) \(\chi_{3600}(1787,\cdot)\) \(\chi_{3600}(2003,\cdot)\) \(\chi_{3600}(2027,\cdot)\) \(\chi_{3600}(2723,\cdot)\) \(\chi_{3600}(2747,\cdot)\) \(\chi_{3600}(2963,\cdot)\) \(\chi_{3600}(3227,\cdot)\) \(\chi_{3600}(3467,\cdot)\)

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

\((3151,901,2801,577)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(83, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{14}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(83,a) \;\) at \(\;a = \) e.g. 2