Properties

Label 5265.292
Modulus $5265$
Conductor $5265$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([64,27,45]))
 
pari: [g,chi] = znchar(Mod(292,5265))
 

Basic properties

Modulus: \(5265\)
Conductor: \(5265\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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 5265.ij

\(\chi_{5265}(292,\cdot)\) \(\chi_{5265}(457,\cdot)\) \(\chi_{5265}(553,\cdot)\) \(\chi_{5265}(583,\cdot)\) \(\chi_{5265}(877,\cdot)\) \(\chi_{5265}(1042,\cdot)\) \(\chi_{5265}(1138,\cdot)\) \(\chi_{5265}(1168,\cdot)\) \(\chi_{5265}(1462,\cdot)\) \(\chi_{5265}(1627,\cdot)\) \(\chi_{5265}(1723,\cdot)\) \(\chi_{5265}(1753,\cdot)\) \(\chi_{5265}(2047,\cdot)\) \(\chi_{5265}(2212,\cdot)\) \(\chi_{5265}(2308,\cdot)\) \(\chi_{5265}(2338,\cdot)\) \(\chi_{5265}(2632,\cdot)\) \(\chi_{5265}(2797,\cdot)\) \(\chi_{5265}(2893,\cdot)\) \(\chi_{5265}(2923,\cdot)\) \(\chi_{5265}(3217,\cdot)\) \(\chi_{5265}(3382,\cdot)\) \(\chi_{5265}(3478,\cdot)\) \(\chi_{5265}(3508,\cdot)\) \(\chi_{5265}(3802,\cdot)\) \(\chi_{5265}(3967,\cdot)\) \(\chi_{5265}(4063,\cdot)\) \(\chi_{5265}(4093,\cdot)\) \(\chi_{5265}(4387,\cdot)\) \(\chi_{5265}(4552,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((326,2107,2836)\) → \((e\left(\frac{16}{27}\right),i,e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 5265 }(292, a) \) \(1\)\(1\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{17}{54}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{67}{108}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{95}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5265 }(292,a) \;\) at \(\;a = \) e.g. 2