Properties

Label 4725.3296
Modulus $4725$
Conductor $4725$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,54,45]))
 
pari: [g,chi] = znchar(Mod(3296,4725))
 

Basic properties

Modulus: \(4725\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.gm

\(\chi_{4725}(41,\cdot)\) \(\chi_{4725}(146,\cdot)\) \(\chi_{4725}(356,\cdot)\) \(\chi_{4725}(461,\cdot)\) \(\chi_{4725}(671,\cdot)\) \(\chi_{4725}(986,\cdot)\) \(\chi_{4725}(1091,\cdot)\) \(\chi_{4725}(1406,\cdot)\) \(\chi_{4725}(1616,\cdot)\) \(\chi_{4725}(1721,\cdot)\) \(\chi_{4725}(1931,\cdot)\) \(\chi_{4725}(2036,\cdot)\) \(\chi_{4725}(2246,\cdot)\) \(\chi_{4725}(2561,\cdot)\) \(\chi_{4725}(2666,\cdot)\) \(\chi_{4725}(2981,\cdot)\) \(\chi_{4725}(3191,\cdot)\) \(\chi_{4725}(3296,\cdot)\) \(\chi_{4725}(3506,\cdot)\) \(\chi_{4725}(3611,\cdot)\) \(\chi_{4725}(3821,\cdot)\) \(\chi_{4725}(4136,\cdot)\) \(\chi_{4725}(4241,\cdot)\) \(\chi_{4725}(4556,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((4376,1702,2026)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{3}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 4725 }(3296, a) \) \(1\)\(1\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{19}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4725 }(3296,a) \;\) at \(\;a = \) e.g. 2