Properties

Label 8027.33
Modulus $8027$
Conductor $8027$
Order $3828$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(3828))
 
M = H._module
 
chi = DirichletCharacter(H, M([522,517]))
 
pari: [g,chi] = znchar(Mod(33,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3828\)
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 8027.bu

\(\chi_{8027}(7,\cdot)\) \(\chi_{8027}(30,\cdot)\) \(\chi_{8027}(33,\cdot)\) \(\chi_{8027}(34,\cdot)\) \(\chi_{8027}(40,\cdot)\) \(\chi_{8027}(43,\cdot)\) \(\chi_{8027}(44,\cdot)\) \(\chi_{8027}(63,\cdot)\) \(\chi_{8027}(74,\cdot)\) \(\chi_{8027}(84,\cdot)\) \(\chi_{8027}(89,\cdot)\) \(\chi_{8027}(90,\cdot)\) \(\chi_{8027}(97,\cdot)\) \(\chi_{8027}(99,\cdot)\) \(\chi_{8027}(107,\cdot)\) \(\chi_{8027}(112,\cdot)\) \(\chi_{8027}(113,\cdot)\) \(\chi_{8027}(120,\cdot)\) \(\chi_{8027}(129,\cdot)\) \(\chi_{8027}(132,\cdot)\) \(\chi_{8027}(134,\cdot)\) \(\chi_{8027}(149,\cdot)\) \(\chi_{8027}(152,\cdot)\) \(\chi_{8027}(159,\cdot)\) \(\chi_{8027}(166,\cdot)\) \(\chi_{8027}(172,\cdot)\) \(\chi_{8027}(175,\cdot)\) \(\chi_{8027}(176,\cdot)\) \(\chi_{8027}(195,\cdot)\) \(\chi_{8027}(199,\cdot)\) ...

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: $\Q(\zeta_{3828})$
Fixed field: Number field defined by a degree 3828 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{3}{22}\right),e\left(\frac{47}{348}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(33, a) \) \(1\)\(1\)\(e\left(\frac{1561}{3828}\right)\)\(e\left(\frac{1327}{1914}\right)\)\(e\left(\frac{1561}{1914}\right)\)\(e\left(\frac{521}{957}\right)\)\(e\left(\frac{129}{1276}\right)\)\(e\left(\frac{2449}{3828}\right)\)\(e\left(\frac{285}{1276}\right)\)\(e\left(\frac{370}{957}\right)\)\(e\left(\frac{1215}{1276}\right)\)\(e\left(\frac{81}{1276}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(33,a) \;\) at \(\;a = \) e.g. 2