Properties

Label 4003.292
Modulus $4003$
Conductor $4003$
Order $87$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(174))
 
M = H._module
 
chi = DirichletCharacter(H, M([142]))
 
pari: [g,chi] = znchar(Mod(292,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(87\)
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 4003.j

\(\chi_{4003}(10,\cdot)\) \(\chi_{4003}(100,\cdot)\) \(\chi_{4003}(117,\cdot)\) \(\chi_{4003}(127,\cdot)\) \(\chi_{4003}(208,\cdot)\) \(\chi_{4003}(214,\cdot)\) \(\chi_{4003}(285,\cdot)\) \(\chi_{4003}(292,\cdot)\) \(\chi_{4003}(316,\cdot)\) \(\chi_{4003}(318,\cdot)\) \(\chi_{4003}(356,\cdot)\) \(\chi_{4003}(412,\cdot)\) \(\chi_{4003}(479,\cdot)\) \(\chi_{4003}(506,\cdot)\) \(\chi_{4003}(785,\cdot)\) \(\chi_{4003}(787,\cdot)\) \(\chi_{4003}(788,\cdot)\) \(\chi_{4003}(913,\cdot)\) \(\chi_{4003}(945,\cdot)\) \(\chi_{4003}(1020,\cdot)\) \(\chi_{4003}(1049,\cdot)\) \(\chi_{4003}(1057,\cdot)\) \(\chi_{4003}(1165,\cdot)\) \(\chi_{4003}(1201,\cdot)\) \(\chi_{4003}(1270,\cdot)\) \(\chi_{4003}(1321,\cdot)\) \(\chi_{4003}(1385,\cdot)\) \(\chi_{4003}(1444,\cdot)\) \(\chi_{4003}(1618,\cdot)\) \(\chi_{4003}(1622,\cdot)\) ...

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: $\Q(\zeta_{87})$
Fixed field: Number field defined by a degree 87 polynomial

Values on generators

\(2\) → \(e\left(\frac{71}{87}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(292, a) \) \(1\)\(1\)\(e\left(\frac{71}{87}\right)\)\(e\left(\frac{43}{87}\right)\)\(e\left(\frac{55}{87}\right)\)\(e\left(\frac{41}{87}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{47}{87}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{86}{87}\right)\)\(e\left(\frac{25}{87}\right)\)\(e\left(\frac{27}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(292,a) \;\) at \(\;a = \) e.g. 2