Properties

Label 8041.7195
Modulus $8041$
Conductor $731$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,40]))
 
pari: [g,chi] = znchar(Mod(7195,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{731}(616,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.ed

\(\chi_{8041}(353,\cdot)\) \(\chi_{8041}(540,\cdot)\) \(\chi_{8041}(1475,\cdot)\) \(\chi_{8041}(1772,\cdot)\) \(\chi_{8041}(1959,\cdot)\) \(\chi_{8041}(2036,\cdot)\) \(\chi_{8041}(2597,\cdot)\) \(\chi_{8041}(2894,\cdot)\) \(\chi_{8041}(3455,\cdot)\) \(\chi_{8041}(4016,\cdot)\) \(\chi_{8041}(4280,\cdot)\) \(\chi_{8041}(4467,\cdot)\) \(\chi_{8041}(4654,\cdot)\) \(\chi_{8041}(4841,\cdot)\) \(\chi_{8041}(5028,\cdot)\) \(\chi_{8041}(5699,\cdot)\) \(\chi_{8041}(5776,\cdot)\) \(\chi_{8041}(5886,\cdot)\) \(\chi_{8041}(6073,\cdot)\) \(\chi_{8041}(6260,\cdot)\) \(\chi_{8041}(6447,\cdot)\) \(\chi_{8041}(6524,\cdot)\) \(\chi_{8041}(7195,\cdot)\) \(\chi_{8041}(7943,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((6580,2366,562)\) → \((1,-i,e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(7195, a) \) \(1\)\(1\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{79}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(7195,a) \;\) at \(\;a = \) e.g. 2