Properties

Label 1375.123
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([51,10]))
 
pari: [g,chi] = znchar(Mod(123,1375))
 

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.cm

\(\chi_{1375}(2,\cdot)\) \(\chi_{1375}(8,\cdot)\) \(\chi_{1375}(123,\cdot)\) \(\chi_{1375}(128,\cdot)\) \(\chi_{1375}(138,\cdot)\) \(\chi_{1375}(172,\cdot)\) \(\chi_{1375}(217,\cdot)\) \(\chi_{1375}(237,\cdot)\) \(\chi_{1375}(277,\cdot)\) \(\chi_{1375}(283,\cdot)\) \(\chi_{1375}(398,\cdot)\) \(\chi_{1375}(403,\cdot)\) \(\chi_{1375}(413,\cdot)\) \(\chi_{1375}(447,\cdot)\) \(\chi_{1375}(492,\cdot)\) \(\chi_{1375}(512,\cdot)\) \(\chi_{1375}(552,\cdot)\) \(\chi_{1375}(558,\cdot)\) \(\chi_{1375}(673,\cdot)\) \(\chi_{1375}(678,\cdot)\) \(\chi_{1375}(688,\cdot)\) \(\chi_{1375}(722,\cdot)\) \(\chi_{1375}(767,\cdot)\) \(\chi_{1375}(787,\cdot)\) \(\chi_{1375}(827,\cdot)\) \(\chi_{1375}(833,\cdot)\) \(\chi_{1375}(948,\cdot)\) \(\chi_{1375}(953,\cdot)\) \(\chi_{1375}(963,\cdot)\) \(\chi_{1375}(997,\cdot)\) ...

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

\((1002,376)\) → \((e\left(\frac{51}{100}\right),e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{37}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{49}{50}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{59}{100}\right)\)\(e\left(\frac{99}{100}\right)\)\(e\left(\frac{33}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(123,a) \;\) at \(\;a = \) e.g. 2