Properties

Label 2583.2
Modulus $2583$
Conductor $2583$
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(2583, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,20,39]))
 
pari: [g,chi] = znchar(Mod(2,2583))
 

Basic properties

Modulus: \(2583\)
Conductor: \(2583\)
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 2583.fg

\(\chi_{2583}(2,\cdot)\) \(\chi_{2583}(128,\cdot)\) \(\chi_{2583}(254,\cdot)\) \(\chi_{2583}(443,\cdot)\) \(\chi_{2583}(569,\cdot)\) \(\chi_{2583}(695,\cdot)\) \(\chi_{2583}(758,\cdot)\) \(\chi_{2583}(1292,\cdot)\) \(\chi_{2583}(1355,\cdot)\) \(\chi_{2583}(1481,\cdot)\) \(\chi_{2583}(1607,\cdot)\) \(\chi_{2583}(1796,\cdot)\) \(\chi_{2583}(1922,\cdot)\) \(\chi_{2583}(2048,\cdot)\) \(\chi_{2583}(2111,\cdot)\) \(\chi_{2583}(2522,\cdot)\)

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

\((2297,2215,1072)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 2583 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{31}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2583 }(2,a) \;\) at \(\;a = \) e.g. 2