Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676000, base_ring=CyclotomicField(300)) M = H._module chi = DirichletCharacter(H, M([150,225,273,50]))
 
Copy content gp:[g,chi] = znchar(Mod(823, 676000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("676000.823");
 

Basic properties

Modulus: \(676000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(26000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(300\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{26000}(20323,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: no
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 676000.bev

\(\chi_{676000}(23,\cdot)\) \(\chi_{676000}(823,\cdot)\) \(\chi_{676000}(2727,\cdot)\) \(\chi_{676000}(3527,\cdot)\) \(\chi_{676000}(27063,\cdot)\) \(\chi_{676000}(27863,\cdot)\) \(\chi_{676000}(29767,\cdot)\) \(\chi_{676000}(30567,\cdot)\) \(\chi_{676000}(54103,\cdot)\) \(\chi_{676000}(54903,\cdot)\) \(\chi_{676000}(83847,\cdot)\) \(\chi_{676000}(84647,\cdot)\) \(\chi_{676000}(108183,\cdot)\) \(\chi_{676000}(108983,\cdot)\) \(\chi_{676000}(110887,\cdot)\) \(\chi_{676000}(111687,\cdot)\) \(\chi_{676000}(135223,\cdot)\) \(\chi_{676000}(136023,\cdot)\) \(\chi_{676000}(137927,\cdot)\) \(\chi_{676000}(138727,\cdot)\) \(\chi_{676000}(162263,\cdot)\) \(\chi_{676000}(163063,\cdot)\) \(\chi_{676000}(164967,\cdot)\) \(\chi_{676000}(165767,\cdot)\) \(\chi_{676000}(189303,\cdot)\) \(\chi_{676000}(190103,\cdot)\) \(\chi_{676000}(219047,\cdot)\) \(\chi_{676000}(219847,\cdot)\) \(\chi_{676000}(243383,\cdot)\) \(\chi_{676000}(244183,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{300})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 300 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((126751,422501,389377,12001)\) → \((-1,-i,e\left(\frac{91}{100}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 676000 }(823, a) \) \(1\)\(1\)\(e\left(\frac{59}{75}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{43}{75}\right)\)\(e\left(\frac{173}{300}\right)\)\(e\left(\frac{229}{300}\right)\)\(e\left(\frac{289}{300}\right)\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{263}{300}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{101}{300}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 676000 }(823,a) \;\) at \(\;a = \) e.g. 2