Properties

Label 1375.6
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([4,45]))
 
pari: [g,chi] = znchar(Mod(6,1375))
 

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.cc

\(\chi_{1375}(6,\cdot)\) \(\chi_{1375}(46,\cdot)\) \(\chi_{1375}(216,\cdot)\) \(\chi_{1375}(261,\cdot)\) \(\chi_{1375}(281,\cdot)\) \(\chi_{1375}(321,\cdot)\) \(\chi_{1375}(491,\cdot)\) \(\chi_{1375}(536,\cdot)\) \(\chi_{1375}(556,\cdot)\) \(\chi_{1375}(596,\cdot)\) \(\chi_{1375}(766,\cdot)\) \(\chi_{1375}(811,\cdot)\) \(\chi_{1375}(831,\cdot)\) \(\chi_{1375}(871,\cdot)\) \(\chi_{1375}(1041,\cdot)\) \(\chi_{1375}(1086,\cdot)\) \(\chi_{1375}(1106,\cdot)\) \(\chi_{1375}(1146,\cdot)\) \(\chi_{1375}(1316,\cdot)\) \(\chi_{1375}(1361,\cdot)\)

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

\((1002,376)\) → \((e\left(\frac{2}{25}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{49}{50}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{2}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(6,a) \;\) at \(\;a = \) e.g. 2