Properties

Label 2500.2399
Modulus $2500$
Conductor $500$
Order $50$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2500, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([25,31]))
 
pari: [g,chi] = znchar(Mod(2399,2500))
 

Basic properties

Modulus: \(2500\)
Conductor: \(500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{500}(279,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2500.n

\(\chi_{2500}(99,\cdot)\) \(\chi_{2500}(199,\cdot)\) \(\chi_{2500}(299,\cdot)\) \(\chi_{2500}(399,\cdot)\) \(\chi_{2500}(599,\cdot)\) \(\chi_{2500}(699,\cdot)\) \(\chi_{2500}(799,\cdot)\) \(\chi_{2500}(899,\cdot)\) \(\chi_{2500}(1099,\cdot)\) \(\chi_{2500}(1199,\cdot)\) \(\chi_{2500}(1299,\cdot)\) \(\chi_{2500}(1399,\cdot)\) \(\chi_{2500}(1599,\cdot)\) \(\chi_{2500}(1699,\cdot)\) \(\chi_{2500}(1799,\cdot)\) \(\chi_{2500}(1899,\cdot)\) \(\chi_{2500}(2099,\cdot)\) \(\chi_{2500}(2199,\cdot)\) \(\chi_{2500}(2299,\cdot)\) \(\chi_{2500}(2399,\cdot)\)

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

\((1251,1877)\) → \((-1,e\left(\frac{31}{50}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2500 }(2399, a) \) \(-1\)\(1\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{13}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2500 }(2399,a) \;\) at \(\;a = \) e.g. 2