Properties

Label 8027.1059
Modulus $8027$
Conductor $349$
Order $87$
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(174))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14]))
 
pari: [g,chi] = znchar(Mod(1059,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(87\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{349}(12,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8027.ba

\(\chi_{8027}(116,\cdot)\) \(\chi_{8027}(185,\cdot)\) \(\chi_{8027}(254,\cdot)\) \(\chi_{8027}(300,\cdot)\) \(\chi_{8027}(346,\cdot)\) \(\chi_{8027}(369,\cdot)\) \(\chi_{8027}(484,\cdot)\) \(\chi_{8027}(507,\cdot)\) \(\chi_{8027}(622,\cdot)\) \(\chi_{8027}(714,\cdot)\) \(\chi_{8027}(783,\cdot)\) \(\chi_{8027}(806,\cdot)\) \(\chi_{8027}(990,\cdot)\) \(\chi_{8027}(1059,\cdot)\) \(\chi_{8027}(1128,\cdot)\) \(\chi_{8027}(1243,\cdot)\) \(\chi_{8027}(1266,\cdot)\) \(\chi_{8027}(1473,\cdot)\) \(\chi_{8027}(1588,\cdot)\) \(\chi_{8027}(1703,\cdot)\) \(\chi_{8027}(2117,\cdot)\) \(\chi_{8027}(2301,\cdot)\) \(\chi_{8027}(2370,\cdot)\) \(\chi_{8027}(2439,\cdot)\) \(\chi_{8027}(2462,\cdot)\) \(\chi_{8027}(2554,\cdot)\) \(\chi_{8027}(2623,\cdot)\) \(\chi_{8027}(2807,\cdot)\) \(\chi_{8027}(3037,\cdot)\) \(\chi_{8027}(3635,\cdot)\) ...

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

\((350,5935)\) → \((1,e\left(\frac{7}{87}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(1059, a) \) \(1\)\(1\)\(e\left(\frac{7}{87}\right)\)\(e\left(\frac{8}{87}\right)\)\(e\left(\frac{14}{87}\right)\)\(e\left(\frac{23}{87}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{34}{87}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{16}{87}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{20}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(1059,a) \;\) at \(\;a = \) e.g. 2