Properties

Label 1375.829
Modulus $1375$
Conductor $1375$
Order $50$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1375, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([41,10]))
 
pari: [g,chi] = znchar(Mod(829,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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 1375.cf

\(\chi_{1375}(4,\cdot)\) \(\chi_{1375}(64,\cdot)\) \(\chi_{1375}(69,\cdot)\) \(\chi_{1375}(159,\cdot)\) \(\chi_{1375}(279,\cdot)\) \(\chi_{1375}(339,\cdot)\) \(\chi_{1375}(344,\cdot)\) \(\chi_{1375}(434,\cdot)\) \(\chi_{1375}(554,\cdot)\) \(\chi_{1375}(614,\cdot)\) \(\chi_{1375}(619,\cdot)\) \(\chi_{1375}(709,\cdot)\) \(\chi_{1375}(829,\cdot)\) \(\chi_{1375}(889,\cdot)\) \(\chi_{1375}(894,\cdot)\) \(\chi_{1375}(984,\cdot)\) \(\chi_{1375}(1104,\cdot)\) \(\chi_{1375}(1164,\cdot)\) \(\chi_{1375}(1169,\cdot)\) \(\chi_{1375}(1259,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((1002,376)\) → \((e\left(\frac{41}{50}\right),e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{3}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(829,a) \;\) at \(\;a = \) e.g. 2