Properties

Label 4725.106
Modulus $4725$
Conductor $675$
Order $45$
Real no
Primitive no
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([50,36,0]))
 
pari: [g,chi] = znchar(Mod(106,4725))
 

Basic properties

Modulus: \(4725\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{675}(106,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4725.fm

\(\chi_{4725}(106,\cdot)\) \(\chi_{4725}(211,\cdot)\) \(\chi_{4725}(421,\cdot)\) \(\chi_{4725}(736,\cdot)\) \(\chi_{4725}(841,\cdot)\) \(\chi_{4725}(1156,\cdot)\) \(\chi_{4725}(1366,\cdot)\) \(\chi_{4725}(1471,\cdot)\) \(\chi_{4725}(1681,\cdot)\) \(\chi_{4725}(1786,\cdot)\) \(\chi_{4725}(1996,\cdot)\) \(\chi_{4725}(2311,\cdot)\) \(\chi_{4725}(2416,\cdot)\) \(\chi_{4725}(2731,\cdot)\) \(\chi_{4725}(2941,\cdot)\) \(\chi_{4725}(3046,\cdot)\) \(\chi_{4725}(3256,\cdot)\) \(\chi_{4725}(3361,\cdot)\) \(\chi_{4725}(3571,\cdot)\) \(\chi_{4725}(3886,\cdot)\) \(\chi_{4725}(3991,\cdot)\) \(\chi_{4725}(4306,\cdot)\) \(\chi_{4725}(4516,\cdot)\) \(\chi_{4725}(4621,\cdot)\)

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 45 polynomial

Values on generators

\((4376,1702,2026)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{2}{5}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 4725 }(106, a) \) \(1\)\(1\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{23}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4725 }(106,a) \;\) at \(\;a = \) e.g. 2