Properties

Label 4725.317
Modulus $4725$
Conductor $4725$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4725, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([70,117,60]))
 
pari: [g,chi] = znchar(Mod(317,4725))
 

Basic properties

Modulus: \(4725\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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 4725.hc

\(\chi_{4725}(2,\cdot)\) \(\chi_{4725}(128,\cdot)\) \(\chi_{4725}(158,\cdot)\) \(\chi_{4725}(317,\cdot)\) \(\chi_{4725}(347,\cdot)\) \(\chi_{4725}(473,\cdot)\) \(\chi_{4725}(662,\cdot)\) \(\chi_{4725}(758,\cdot)\) \(\chi_{4725}(788,\cdot)\) \(\chi_{4725}(947,\cdot)\) \(\chi_{4725}(977,\cdot)\) \(\chi_{4725}(1073,\cdot)\) \(\chi_{4725}(1103,\cdot)\) \(\chi_{4725}(1262,\cdot)\) \(\chi_{4725}(1292,\cdot)\) \(\chi_{4725}(1388,\cdot)\) \(\chi_{4725}(1577,\cdot)\) \(\chi_{4725}(1703,\cdot)\) \(\chi_{4725}(1733,\cdot)\) \(\chi_{4725}(1892,\cdot)\) \(\chi_{4725}(1922,\cdot)\) \(\chi_{4725}(2048,\cdot)\) \(\chi_{4725}(2237,\cdot)\) \(\chi_{4725}(2333,\cdot)\) \(\chi_{4725}(2363,\cdot)\) \(\chi_{4725}(2522,\cdot)\) \(\chi_{4725}(2552,\cdot)\) \(\chi_{4725}(2648,\cdot)\) \(\chi_{4725}(2678,\cdot)\) \(\chi_{4725}(2837,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((4376,1702,2026)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{13}{20}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 4725 }(317, a) \) \(1\)\(1\)\(e\left(\frac{127}{180}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{83}{180}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{89}{180}\right)\)\(e\left(\frac{17}{180}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4725 }(317,a) \;\) at \(\;a = \) e.g. 2