Properties

Label 1375.809
Modulus $1375$
Conductor $1375$
Order $50$
Real no
Primitive yes
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(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17,45]))
 
pari: [g,chi] = znchar(Mod(809,1375))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1375.by

\(\chi_{1375}(19,\cdot)\) \(\chi_{1375}(29,\cdot)\) \(\chi_{1375}(189,\cdot)\) \(\chi_{1375}(259,\cdot)\) \(\chi_{1375}(294,\cdot)\) \(\chi_{1375}(304,\cdot)\) \(\chi_{1375}(464,\cdot)\) \(\chi_{1375}(534,\cdot)\) \(\chi_{1375}(569,\cdot)\) \(\chi_{1375}(579,\cdot)\) \(\chi_{1375}(739,\cdot)\) \(\chi_{1375}(809,\cdot)\) \(\chi_{1375}(844,\cdot)\) \(\chi_{1375}(854,\cdot)\) \(\chi_{1375}(1014,\cdot)\) \(\chi_{1375}(1084,\cdot)\) \(\chi_{1375}(1119,\cdot)\) \(\chi_{1375}(1129,\cdot)\) \(\chi_{1375}(1289,\cdot)\) \(\chi_{1375}(1359,\cdot)\)

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

\((1002,376)\) → \((e\left(\frac{17}{50}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{11}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(809,a) \;\) at \(\;a = \) e.g. 2