Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(228)) M = H._module chi = DirichletCharacter(H, M([57,100]))
 
Copy content gp:[g,chi] = znchar(Mod(7, 9025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9025.7");
 

Basic properties

Modulus: \(9025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1805\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(228\)
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 \(\chi_{1805}(7,\cdot)\)
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: yes
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 9025.bz

\(\chi_{9025}(7,\cdot)\) \(\chi_{9025}(182,\cdot)\) \(\chi_{9025}(368,\cdot)\) \(\chi_{9025}(482,\cdot)\) \(\chi_{9025}(543,\cdot)\) \(\chi_{9025}(657,\cdot)\) \(\chi_{9025}(843,\cdot)\) \(\chi_{9025}(957,\cdot)\) \(\chi_{9025}(1018,\cdot)\) \(\chi_{9025}(1132,\cdot)\) \(\chi_{9025}(1318,\cdot)\) \(\chi_{9025}(1432,\cdot)\) \(\chi_{9025}(1493,\cdot)\) \(\chi_{9025}(1607,\cdot)\) \(\chi_{9025}(1793,\cdot)\) \(\chi_{9025}(1907,\cdot)\) \(\chi_{9025}(1968,\cdot)\) \(\chi_{9025}(2082,\cdot)\) \(\chi_{9025}(2268,\cdot)\) \(\chi_{9025}(2382,\cdot)\) \(\chi_{9025}(2443,\cdot)\) \(\chi_{9025}(2557,\cdot)\) \(\chi_{9025}(2743,\cdot)\) \(\chi_{9025}(2857,\cdot)\) \(\chi_{9025}(2918,\cdot)\) \(\chi_{9025}(3032,\cdot)\) \(\chi_{9025}(3218,\cdot)\) \(\chi_{9025}(3332,\cdot)\) \(\chi_{9025}(3393,\cdot)\) \(\chi_{9025}(3507,\cdot)\) ...

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: $\Q(\zeta_{228})$
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 228 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((5777,3251)\) → \((i,e\left(\frac{25}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 9025 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{157}{228}\right)\)\(e\left(\frac{163}{228}\right)\)\(e\left(\frac{43}{114}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{3}{76}\right)\)\(e\left(\frac{5}{76}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{7}{76}\right)\)\(e\left(\frac{11}{228}\right)\)
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)
 
\( \chi_{ 9025 }(7,a) \;\) at \(\;a = \) e.g. 2