Properties

Label 1375.23
Modulus $1375$
Conductor $125$
Order $100$
Real no
Primitive no
Minimal yes
Parity odd

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([31,0]))
 
pari: [g,chi] = znchar(Mod(23,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1375.cq

\(\chi_{1375}(12,\cdot)\) \(\chi_{1375}(23,\cdot)\) \(\chi_{1375}(67,\cdot)\) \(\chi_{1375}(78,\cdot)\) \(\chi_{1375}(122,\cdot)\) \(\chi_{1375}(133,\cdot)\) \(\chi_{1375}(177,\cdot)\) \(\chi_{1375}(188,\cdot)\) \(\chi_{1375}(287,\cdot)\) \(\chi_{1375}(298,\cdot)\) \(\chi_{1375}(342,\cdot)\) \(\chi_{1375}(353,\cdot)\) \(\chi_{1375}(397,\cdot)\) \(\chi_{1375}(408,\cdot)\) \(\chi_{1375}(452,\cdot)\) \(\chi_{1375}(463,\cdot)\) \(\chi_{1375}(562,\cdot)\) \(\chi_{1375}(573,\cdot)\) \(\chi_{1375}(617,\cdot)\) \(\chi_{1375}(628,\cdot)\) \(\chi_{1375}(672,\cdot)\) \(\chi_{1375}(683,\cdot)\) \(\chi_{1375}(727,\cdot)\) \(\chi_{1375}(738,\cdot)\) \(\chi_{1375}(837,\cdot)\) \(\chi_{1375}(848,\cdot)\) \(\chi_{1375}(892,\cdot)\) \(\chi_{1375}(903,\cdot)\) \(\chi_{1375}(947,\cdot)\) \(\chi_{1375}(958,\cdot)\) ...

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

\((1002,376)\) → \((e\left(\frac{31}{100}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{17}{100}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{33}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(23,a) \;\) at \(\;a = \) e.g. 2