Properties

Label 4033.17
Modulus $4033$
Conductor $4033$
Order $36$
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(4033)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,31]))
 
pari: [g,chi] = znchar(Mod(17,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 4033.gn

\(\chi_{4033}(17,\cdot)\) \(\chi_{4033}(241,\cdot)\) \(\chi_{4033}(708,\cdot)\) \(\chi_{4033}(949,\cdot)\) \(\chi_{4033}(1071,\cdot)\) \(\chi_{4033}(1088,\cdot)\) \(\chi_{4033}(2945,\cdot)\) \(\chi_{4033}(2962,\cdot)\) \(\chi_{4033}(3084,\cdot)\) \(\chi_{4033}(3325,\cdot)\) \(\chi_{4033}(3792,\cdot)\) \(\chi_{4033}(4016,\cdot)\)

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

\((1963,2295)\) → \((e\left(\frac{7}{36}\right),e\left(\frac{31}{36}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{11}{36}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial