Properties

Label 1205.493
Modulus $1205$
Conductor $1205$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,5]))
 
pari: [g,chi] = znchar(Mod(493,1205))
 

Basic properties

Modulus: \(1205\)
Conductor: \(1205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1205.cb

\(\chi_{1205}(22,\cdot)\) \(\chi_{1205}(63,\cdot)\) \(\chi_{1205}(152,\cdot)\) \(\chi_{1205}(178,\cdot)\) \(\chi_{1205}(463,\cdot)\) \(\chi_{1205}(493,\cdot)\) \(\chi_{1205}(658,\cdot)\) \(\chi_{1205}(788,\cdot)\) \(\chi_{1205}(812,\cdot)\) \(\chi_{1205}(942,\cdot)\) \(\chi_{1205}(953,\cdot)\) \(\chi_{1205}(983,\cdot)\) \(\chi_{1205}(1002,\cdot)\) \(\chi_{1205}(1052,\cdot)\) \(\chi_{1205}(1117,\cdot)\) \(\chi_{1205}(1167,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((242,971)\) → \((-i,e\left(\frac{5}{48}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1205 }(493, a) \) \(1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(-i\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{7}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1205 }(493,a) \;\) at \(\;a = \) e.g. 2