Properties

Label 8041.2393
Modulus $8041$
Conductor $8041$
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(8041, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([378,105,50]))
 
pari: [g,chi] = znchar(Mod(2393,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
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 8041.fp

\(\chi_{8041}(30,\cdot)\) \(\chi_{8041}(72,\cdot)\) \(\chi_{8041}(106,\cdot)\) \(\chi_{8041}(149,\cdot)\) \(\chi_{8041}(200,\cdot)\) \(\chi_{8041}(327,\cdot)\) \(\chi_{8041}(370,\cdot)\) \(\chi_{8041}(761,\cdot)\) \(\chi_{8041}(820,\cdot)\) \(\chi_{8041}(888,\cdot)\) \(\chi_{8041}(931,\cdot)\) \(\chi_{8041}(965,\cdot)\) \(\chi_{8041}(1007,\cdot)\) \(\chi_{8041}(1058,\cdot)\) \(\chi_{8041}(1152,\cdot)\) \(\chi_{8041}(1194,\cdot)\) \(\chi_{8041}(1381,\cdot)\) \(\chi_{8041}(1449,\cdot)\) \(\chi_{8041}(1492,\cdot)\) \(\chi_{8041}(1568,\cdot)\) \(\chi_{8041}(1619,\cdot)\) \(\chi_{8041}(1696,\cdot)\) \(\chi_{8041}(1789,\cdot)\) \(\chi_{8041}(1832,\cdot)\) \(\chi_{8041}(1883,\cdot)\) \(\chi_{8041}(2180,\cdot)\) \(\chi_{8041}(2350,\cdot)\) \(\chi_{8041}(2384,\cdot)\) \(\chi_{8041}(2393,\cdot)\) \(\chi_{8041}(2427,\cdot)\) ...

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

\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),i,e\left(\frac{5}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(2393, a) \) \(1\)\(1\)\(e\left(\frac{43}{70}\right)\)\(e\left(\frac{239}{420}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{347}{420}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{29}{210}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{67}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(2393,a) \;\) at \(\;a = \) e.g. 2