Properties

Label 1375.28
Modulus $1375$
Conductor $1375$
Order $100$
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(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([87,90]))
 
pari: [g,chi] = znchar(Mod(28,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
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.cl

\(\chi_{1375}(28,\cdot)\) \(\chi_{1375}(63,\cdot)\) \(\chi_{1375}(72,\cdot)\) \(\chi_{1375}(73,\cdot)\) \(\chi_{1375}(162,\cdot)\) \(\chi_{1375}(167,\cdot)\) \(\chi_{1375}(227,\cdot)\) \(\chi_{1375}(233,\cdot)\) \(\chi_{1375}(303,\cdot)\) \(\chi_{1375}(338,\cdot)\) \(\chi_{1375}(347,\cdot)\) \(\chi_{1375}(348,\cdot)\) \(\chi_{1375}(437,\cdot)\) \(\chi_{1375}(442,\cdot)\) \(\chi_{1375}(502,\cdot)\) \(\chi_{1375}(508,\cdot)\) \(\chi_{1375}(578,\cdot)\) \(\chi_{1375}(613,\cdot)\) \(\chi_{1375}(622,\cdot)\) \(\chi_{1375}(623,\cdot)\) \(\chi_{1375}(712,\cdot)\) \(\chi_{1375}(717,\cdot)\) \(\chi_{1375}(777,\cdot)\) \(\chi_{1375}(783,\cdot)\) \(\chi_{1375}(853,\cdot)\) \(\chi_{1375}(888,\cdot)\) \(\chi_{1375}(897,\cdot)\) \(\chi_{1375}(898,\cdot)\) \(\chi_{1375}(987,\cdot)\) \(\chi_{1375}(992,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((1002,376)\) → \((e\left(\frac{87}{100}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{77}{100}\right)\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{3}{50}\right)\)\(i\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{1}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(28,a) \;\) at \(\;a = \) e.g. 2