Properties

Label 1375.34
Modulus $1375$
Conductor $125$
Order $50$
Real no
Primitive no
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(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([37,0]))
 
pari: [g,chi] = znchar(Mod(34,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1375.ce

\(\chi_{1375}(34,\cdot)\) \(\chi_{1375}(89,\cdot)\) \(\chi_{1375}(144,\cdot)\) \(\chi_{1375}(254,\cdot)\) \(\chi_{1375}(309,\cdot)\) \(\chi_{1375}(364,\cdot)\) \(\chi_{1375}(419,\cdot)\) \(\chi_{1375}(529,\cdot)\) \(\chi_{1375}(584,\cdot)\) \(\chi_{1375}(639,\cdot)\) \(\chi_{1375}(694,\cdot)\) \(\chi_{1375}(804,\cdot)\) \(\chi_{1375}(859,\cdot)\) \(\chi_{1375}(914,\cdot)\) \(\chi_{1375}(969,\cdot)\) \(\chi_{1375}(1079,\cdot)\) \(\chi_{1375}(1134,\cdot)\) \(\chi_{1375}(1189,\cdot)\) \(\chi_{1375}(1244,\cdot)\) \(\chi_{1375}(1354,\cdot)\)

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

\((1002,376)\) → \((e\left(\frac{37}{50}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{16}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(34,a) \;\) at \(\;a = \) e.g. 2