Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4560, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([0,0,18,9,8]))
 
Copy content gp:[g,chi] = znchar(Mod(2657, 4560))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4560.2657");
 

Basic properties

Modulus: \(4560\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(285\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
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 \(\chi_{285}(92,\cdot)\)
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 4560.iq

\(\chi_{4560}(17,\cdot)\) \(\chi_{4560}(593,\cdot)\) \(\chi_{4560}(833,\cdot)\) \(\chi_{4560}(1073,\cdot)\) \(\chi_{4560}(1697,\cdot)\) \(\chi_{4560}(2417,\cdot)\) \(\chi_{4560}(2513,\cdot)\) \(\chi_{4560}(2657,\cdot)\) \(\chi_{4560}(2753,\cdot)\) \(\chi_{4560}(2897,\cdot)\) \(\chi_{4560}(4337,\cdot)\) \(\chi_{4560}(4433,\cdot)\)

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: \(\Q(\zeta_{36})\)
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 36 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1711,1141,3041,2737,1921)\) → \((1,1,-1,i,e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4560 }(2657, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{36}\right)\)
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)
 
\( \chi_{ 4560 }(2657,a) \;\) at \(\;a = \) e.g. 2