Properties

Label 4900.23
Modulus $4900$
Conductor $4900$
Order $420$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([210,231,380]))
 
pari: [g,chi] = znchar(Mod(23,4900))
 

Basic properties

Modulus: \(4900\)
Conductor: \(4900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
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 4900.dr

\(\chi_{4900}(23,\cdot)\) \(\chi_{4900}(123,\cdot)\) \(\chi_{4900}(163,\cdot)\) \(\chi_{4900}(247,\cdot)\) \(\chi_{4900}(303,\cdot)\) \(\chi_{4900}(347,\cdot)\) \(\chi_{4900}(387,\cdot)\) \(\chi_{4900}(403,\cdot)\) \(\chi_{4900}(487,\cdot)\) \(\chi_{4900}(527,\cdot)\) \(\chi_{4900}(583,\cdot)\) \(\chi_{4900}(627,\cdot)\) \(\chi_{4900}(683,\cdot)\) \(\chi_{4900}(723,\cdot)\) \(\chi_{4900}(767,\cdot)\) \(\chi_{4900}(823,\cdot)\) \(\chi_{4900}(947,\cdot)\) \(\chi_{4900}(963,\cdot)\) \(\chi_{4900}(1003,\cdot)\) \(\chi_{4900}(1087,\cdot)\) \(\chi_{4900}(1103,\cdot)\) \(\chi_{4900}(1187,\cdot)\) \(\chi_{4900}(1227,\cdot)\) \(\chi_{4900}(1283,\cdot)\) \(\chi_{4900}(1327,\cdot)\) \(\chi_{4900}(1367,\cdot)\) \(\chi_{4900}(1383,\cdot)\) \(\chi_{4900}(1423,\cdot)\) \(\chi_{4900}(1467,\cdot)\) \(\chi_{4900}(1523,\cdot)\)

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((2451,1177,101)\) → \((-1,e\left(\frac{11}{20}\right),e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 4900 }(23, a) \) \(1\)\(1\)\(e\left(\frac{107}{420}\right)\)\(e\left(\frac{107}{210}\right)\)\(e\left(\frac{103}{210}\right)\)\(e\left(\frac{43}{140}\right)\)\(e\left(\frac{323}{420}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{391}{420}\right)\)\(e\left(\frac{107}{140}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4900 }(23,a) \;\) at \(\;a = \) e.g. 2