Properties

Label 8027.17
Modulus $8027$
Conductor $8027$
Order $638$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(638))
 
M = H._module
 
chi = DirichletCharacter(H, M([203,473]))
 
pari: [g,chi] = znchar(Mod(17,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(638\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8027.bl

\(\chi_{8027}(17,\cdot)\) \(\chi_{8027}(37,\cdot)\) \(\chi_{8027}(60,\cdot)\) \(\chi_{8027}(80,\cdot)\) \(\chi_{8027}(86,\cdot)\) \(\chi_{8027}(125,\cdot)\) \(\chi_{8027}(178,\cdot)\) \(\chi_{8027}(181,\cdot)\) \(\chi_{8027}(283,\cdot)\) \(\chi_{8027}(318,\cdot)\) \(\chi_{8027}(366,\cdot)\) \(\chi_{8027}(385,\cdot)\) \(\chi_{8027}(424,\cdot)\) \(\chi_{8027}(429,\cdot)\) \(\chi_{8027}(435,\cdot)\) \(\chi_{8027}(470,\cdot)\) \(\chi_{8027}(474,\cdot)\) \(\chi_{8027}(488,\cdot)\) \(\chi_{8027}(527,\cdot)\) \(\chi_{8027}(572,\cdot)\) \(\chi_{8027}(580,\cdot)\) \(\chi_{8027}(631,\cdot)\) \(\chi_{8027}(632,\cdot)\) \(\chi_{8027}(734,\cdot)\) \(\chi_{8027}(743,\cdot)\) \(\chi_{8027}(746,\cdot)\) \(\chi_{8027}(773,\cdot)\) \(\chi_{8027}(778,\cdot)\) \(\chi_{8027}(819,\cdot)\) \(\chi_{8027}(879,\cdot)\) ...

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: $\Q(\zeta_{319})$
Fixed field: Number field defined by a degree 638 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{7}{22}\right),e\left(\frac{43}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{241}{638}\right)\)\(e\left(\frac{117}{319}\right)\)\(e\left(\frac{241}{319}\right)\)\(e\left(\frac{71}{638}\right)\)\(e\left(\frac{475}{638}\right)\)\(e\left(\frac{229}{319}\right)\)\(e\left(\frac{85}{638}\right)\)\(e\left(\frac{234}{319}\right)\)\(e\left(\frac{156}{319}\right)\)\(e\left(\frac{138}{319}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(17,a) \;\) at \(\;a = \) e.g. 2