Properties

Label 8033.37
Modulus $8033$
Conductor $8033$
Order $644$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8033)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([69,203]))
 
pari: [g,chi] = znchar(Mod(37,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(644\)
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 8033.cn

\(\chi_{8033}(37,\cdot)\) \(\chi_{8033}(73,\cdot)\) \(\chi_{8033}(118,\cdot)\) \(\chi_{8033}(148,\cdot)\) \(\chi_{8033}(269,\cdot)\) \(\chi_{8033}(275,\cdot)\) \(\chi_{8033}(279,\cdot)\) \(\chi_{8033}(292,\cdot)\) \(\chi_{8033}(359,\cdot)\) \(\chi_{8033}(409,\cdot)\) \(\chi_{8033}(425,\cdot)\) \(\chi_{8033}(445,\cdot)\) \(\chi_{8033}(472,\cdot)\) \(\chi_{8033}(562,\cdot)\) \(\chi_{8033}(627,\cdot)\) \(\chi_{8033}(686,\cdot)\) \(\chi_{8033}(722,\cdot)\) \(\chi_{8033}(727,\cdot)\) \(\chi_{8033}(794,\cdot)\) \(\chi_{8033}(868,\cdot)\) \(\chi_{8033}(873,\cdot)\) \(\chi_{8033}(885,\cdot)\) \(\chi_{8033}(913,\cdot)\) \(\chi_{8033}(949,\cdot)\) \(\chi_{8033}(959,\cdot)\) \(\chi_{8033}(1047,\cdot)\) \(\chi_{8033}(1075,\cdot)\) \(\chi_{8033}(1076,\cdot)\) \(\chi_{8033}(1100,\cdot)\) \(\chi_{8033}(1110,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((5541,1944)\) → \((e\left(\frac{3}{28}\right),e\left(\frac{29}{92}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{143}{322}\right)\)\(e\left(\frac{513}{644}\right)\)\(e\left(\frac{143}{161}\right)\)\(e\left(\frac{433}{644}\right)\)\(e\left(\frac{155}{644}\right)\)\(e\left(\frac{71}{322}\right)\)\(e\left(\frac{107}{322}\right)\)\(e\left(\frac{191}{322}\right)\)\(e\left(\frac{75}{644}\right)\)\(e\left(\frac{285}{322}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{644})$
Fixed field: Number field defined by a degree 644 polynomial