Properties

Label 8027.1979
Modulus $8027$
Conductor $349$
Order $29$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,54]))
 
pari: [g,chi] = znchar(Mod(1979,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{349}(234,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8027.q

\(\chi_{8027}(415,\cdot)\) \(\chi_{8027}(967,\cdot)\) \(\chi_{8027}(1427,\cdot)\) \(\chi_{8027}(1979,\cdot)\) \(\chi_{8027}(2002,\cdot)\) \(\chi_{8027}(2025,\cdot)\) \(\chi_{8027}(2416,\cdot)\) \(\chi_{8027}(2531,\cdot)\) \(\chi_{8027}(2692,\cdot)\) \(\chi_{8027}(3267,\cdot)\) \(\chi_{8027}(3658,\cdot)\) \(\chi_{8027}(3957,\cdot)\) \(\chi_{8027}(4049,\cdot)\) \(\chi_{8027}(4578,\cdot)\) \(\chi_{8027}(4647,\cdot)\) \(\chi_{8027}(5199,\cdot)\) \(\chi_{8027}(5406,\cdot)\) \(\chi_{8027}(5498,\cdot)\) \(\chi_{8027}(5567,\cdot)\) \(\chi_{8027}(6234,\cdot)\) \(\chi_{8027}(6349,\cdot)\) \(\chi_{8027}(6510,\cdot)\) \(\chi_{8027}(6556,\cdot)\) \(\chi_{8027}(6855,\cdot)\) \(\chi_{8027}(7269,\cdot)\) \(\chi_{8027}(7292,\cdot)\) \(\chi_{8027}(7614,\cdot)\) \(\chi_{8027}(7982,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 29 polynomial

Values on generators

\((350,5935)\) → \((1,e\left(\frac{27}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(1979, a) \) \(1\)\(1\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{16}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(1979,a) \;\) at \(\;a = \) e.g. 2