Properties

Label 4725.52
Modulus $4725$
Conductor $4725$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4725, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([100,9,30]))
 
pari: [g,chi] = znchar(Mod(52,4725))
 

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.gz

\(\chi_{4725}(52,\cdot)\) \(\chi_{4725}(103,\cdot)\) \(\chi_{4725}(178,\cdot)\) \(\chi_{4725}(292,\cdot)\) \(\chi_{4725}(367,\cdot)\) \(\chi_{4725}(733,\cdot)\) \(\chi_{4725}(808,\cdot)\) \(\chi_{4725}(922,\cdot)\) \(\chi_{4725}(997,\cdot)\) \(\chi_{4725}(1048,\cdot)\) \(\chi_{4725}(1123,\cdot)\) \(\chi_{4725}(1237,\cdot)\) \(\chi_{4725}(1312,\cdot)\) \(\chi_{4725}(1363,\cdot)\) \(\chi_{4725}(1438,\cdot)\) \(\chi_{4725}(1552,\cdot)\) \(\chi_{4725}(1627,\cdot)\) \(\chi_{4725}(1678,\cdot)\) \(\chi_{4725}(1753,\cdot)\) \(\chi_{4725}(1867,\cdot)\) \(\chi_{4725}(1942,\cdot)\) \(\chi_{4725}(2308,\cdot)\) \(\chi_{4725}(2383,\cdot)\) \(\chi_{4725}(2497,\cdot)\) \(\chi_{4725}(2572,\cdot)\) \(\chi_{4725}(2623,\cdot)\) \(\chi_{4725}(2698,\cdot)\) \(\chi_{4725}(2812,\cdot)\) \(\chi_{4725}(2887,\cdot)\) \(\chi_{4725}(2938,\cdot)\) ...

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

\((4376,1702,2026)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 4725 }(52, a) \) \(1\)\(1\)\(e\left(\frac{169}{180}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{161}{180}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{113}{180}\right)\)\(e\left(\frac{179}{180}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4725 }(52,a) \;\) at \(\;a = \) e.g. 2