Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(221184, base_ring=CyclotomicField(2304)) M = H._module chi = DirichletCharacter(H, M([1152,1035,2176]))
 
Copy content gp:[g,chi] = znchar(Mod(95, 221184))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("221184.95");
 

Basic properties

Modulus: \(221184\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(27648\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2304\)
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 \(\chi_{27648}(27203,\cdot)\)
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 221184.eq

\(\chi_{221184}(95,\cdot)\) \(\chi_{221184}(479,\cdot)\) \(\chi_{221184}(671,\cdot)\) \(\chi_{221184}(1055,\cdot)\) \(\chi_{221184}(1247,\cdot)\) \(\chi_{221184}(1631,\cdot)\) \(\chi_{221184}(1823,\cdot)\) \(\chi_{221184}(2207,\cdot)\) \(\chi_{221184}(2399,\cdot)\) \(\chi_{221184}(2783,\cdot)\) \(\chi_{221184}(2975,\cdot)\) \(\chi_{221184}(3359,\cdot)\) \(\chi_{221184}(3551,\cdot)\) \(\chi_{221184}(3935,\cdot)\) \(\chi_{221184}(4127,\cdot)\) \(\chi_{221184}(4511,\cdot)\) \(\chi_{221184}(4703,\cdot)\) \(\chi_{221184}(5087,\cdot)\) \(\chi_{221184}(5279,\cdot)\) \(\chi_{221184}(5663,\cdot)\) \(\chi_{221184}(5855,\cdot)\) \(\chi_{221184}(6239,\cdot)\) \(\chi_{221184}(6431,\cdot)\) \(\chi_{221184}(6815,\cdot)\) \(\chi_{221184}(7007,\cdot)\) \(\chi_{221184}(7391,\cdot)\) \(\chi_{221184}(7583,\cdot)\) \(\chi_{221184}(7967,\cdot)\) \(\chi_{221184}(8159,\cdot)\) \(\chi_{221184}(8543,\cdot)\) ...

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: $\Q(\zeta_{2304})$
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 2304 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((81919,57349,40961)\) → \((-1,e\left(\frac{115}{256}\right),e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 221184 }(95, a) \) \(1\)\(1\)\(e\left(\frac{395}{2304}\right)\)\(e\left(\frac{407}{1152}\right)\)\(e\left(\frac{1063}{2304}\right)\)\(e\left(\frac{2117}{2304}\right)\)\(e\left(\frac{47}{192}\right)\)\(e\left(\frac{511}{768}\right)\)\(e\left(\frac{781}{1152}\right)\)\(e\left(\frac{395}{1152}\right)\)\(e\left(\frac{457}{2304}\right)\)\(e\left(\frac{139}{288}\right)\)
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)
 
\( \chi_{ 221184 }(95,a) \;\) at \(\;a = \) e.g. 2