Properties

Label 1875.4
Modulus $1875$
Conductor $625$
Order $250$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(4,1875))
 

Basic properties

Modulus: \(1875\)
Conductor: \(625\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{625}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1875.v

\(\chi_{1875}(4,\cdot)\) \(\chi_{1875}(19,\cdot)\) \(\chi_{1875}(34,\cdot)\) \(\chi_{1875}(64,\cdot)\) \(\chi_{1875}(79,\cdot)\) \(\chi_{1875}(94,\cdot)\) \(\chi_{1875}(109,\cdot)\) \(\chi_{1875}(139,\cdot)\) \(\chi_{1875}(154,\cdot)\) \(\chi_{1875}(169,\cdot)\) \(\chi_{1875}(184,\cdot)\) \(\chi_{1875}(214,\cdot)\) \(\chi_{1875}(229,\cdot)\) \(\chi_{1875}(244,\cdot)\) \(\chi_{1875}(259,\cdot)\) \(\chi_{1875}(289,\cdot)\) \(\chi_{1875}(304,\cdot)\) \(\chi_{1875}(319,\cdot)\) \(\chi_{1875}(334,\cdot)\) \(\chi_{1875}(364,\cdot)\) \(\chi_{1875}(379,\cdot)\) \(\chi_{1875}(394,\cdot)\) \(\chi_{1875}(409,\cdot)\) \(\chi_{1875}(439,\cdot)\) \(\chi_{1875}(454,\cdot)\) \(\chi_{1875}(469,\cdot)\) \(\chi_{1875}(484,\cdot)\) \(\chi_{1875}(514,\cdot)\) \(\chi_{1875}(529,\cdot)\) \(\chi_{1875}(544,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((626,1252)\) → \((1,e\left(\frac{1}{250}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1875 }(4, a) \) \(1\)\(1\)\(e\left(\frac{1}{250}\right)\)\(e\left(\frac{1}{125}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{3}{250}\right)\)\(e\left(\frac{113}{125}\right)\)\(e\left(\frac{139}{250}\right)\)\(e\left(\frac{118}{125}\right)\)\(e\left(\frac{2}{125}\right)\)\(e\left(\frac{173}{250}\right)\)\(e\left(\frac{84}{125}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1875 }(4,a) \;\) at \(\;a = \) e.g. 2