Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([0,81,50]))
 
Copy content gp:[g,chi] = znchar(Mod(425, 2592))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.425");
 

Basic properties

Modulus: \(2592\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1296\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(108\)
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 \(\chi_{1296}(749,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2592.cg

\(\chi_{2592}(41,\cdot)\) \(\chi_{2592}(137,\cdot)\) \(\chi_{2592}(185,\cdot)\) \(\chi_{2592}(281,\cdot)\) \(\chi_{2592}(329,\cdot)\) \(\chi_{2592}(425,\cdot)\) \(\chi_{2592}(473,\cdot)\) \(\chi_{2592}(569,\cdot)\) \(\chi_{2592}(617,\cdot)\) \(\chi_{2592}(713,\cdot)\) \(\chi_{2592}(761,\cdot)\) \(\chi_{2592}(857,\cdot)\) \(\chi_{2592}(905,\cdot)\) \(\chi_{2592}(1001,\cdot)\) \(\chi_{2592}(1049,\cdot)\) \(\chi_{2592}(1145,\cdot)\) \(\chi_{2592}(1193,\cdot)\) \(\chi_{2592}(1289,\cdot)\) \(\chi_{2592}(1337,\cdot)\) \(\chi_{2592}(1433,\cdot)\) \(\chi_{2592}(1481,\cdot)\) \(\chi_{2592}(1577,\cdot)\) \(\chi_{2592}(1625,\cdot)\) \(\chi_{2592}(1721,\cdot)\) \(\chi_{2592}(1769,\cdot)\) \(\chi_{2592}(1865,\cdot)\) \(\chi_{2592}(1913,\cdot)\) \(\chi_{2592}(2009,\cdot)\) \(\chi_{2592}(2057,\cdot)\) \(\chi_{2592}(2153,\cdot)\) ...

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: $\Q(\zeta_{108})$
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 108 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((2431,325,1217)\) → \((1,-i,e\left(\frac{25}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2592 }(425, a) \) \(-1\)\(1\)\(e\left(\frac{43}{108}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{83}{108}\right)\)\(e\left(\frac{103}{108}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{41}{108}\right)\)\(e\left(\frac{7}{27}\right)\)
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)
 
\( \chi_{ 2592 }(425,a) \;\) at \(\;a = \) e.g. 2