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Properties

Label 388075.23203
Modulus $388075$
Conductor $20425$
Order $1260$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(388075, base_ring=CyclotomicField(1260)) M = H._module chi = DirichletCharacter(H, M([441,140,510]))
 
Copy content gp:[g,chi] = znchar(Mod(23203, 388075))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("388075.23203");
 

Basic properties

Modulus: \(388075\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(20425\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1260\)
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 \(\chi_{20425}(2778,\cdot)\)
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 388075.wm

\(\chi_{388075}(28,\cdot)\) \(\chi_{388075}(62,\cdot)\) \(\chi_{388075}(1137,\cdot)\) \(\chi_{388075}(2772,\cdot)\) \(\chi_{388075}(3122,\cdot)\) \(\chi_{388075}(3638,\cdot)\) \(\chi_{388075}(4577,\cdot)\) \(\chi_{388075}(5838,\cdot)\) \(\chi_{388075}(6597,\cdot)\) \(\chi_{388075}(6913,\cdot)\) \(\chi_{388075}(8763,\cdot)\) \(\chi_{388075}(10353,\cdot)\) \(\chi_{388075}(12697,\cdot)\) \(\chi_{388075}(13772,\cdot)\) \(\chi_{388075}(15983,\cdot)\) \(\chi_{388075}(17212,\cdot)\) \(\chi_{388075}(17562,\cdot)\) \(\chi_{388075}(17788,\cdot)\) \(\chi_{388075}(18078,\cdot)\) \(\chi_{388075}(19728,\cdot)\) \(\chi_{388075}(23203,\cdot)\) \(\chi_{388075}(26948,\cdot)\) \(\chi_{388075}(28753,\cdot)\) \(\chi_{388075}(28942,\cdot)\) \(\chi_{388075}(29303,\cdot)\) \(\chi_{388075}(29847,\cdot)\) \(\chi_{388075}(30017,\cdot)\) \(\chi_{388075}(30378,\cdot)\) \(\chi_{388075}(30558,\cdot)\) \(\chi_{388075}(31108,\cdot)\) ...

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: $\Q(\zeta_{1260})$
Fixed field: Number field defined by a degree 1260 polynomial (not computed)

Values on generators

\((186277,48376,306851)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{1}{9}\right),e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 388075 }(23203, a) \) \(1\)\(1\)\(e\left(\frac{491}{1260}\right)\)\(e\left(\frac{377}{1260}\right)\)\(e\left(\frac{491}{630}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{71}{420}\right)\)\(e\left(\frac{377}{630}\right)\)\(e\left(\frac{8}{105}\right)\)\(e\left(\frac{11}{140}\right)\)\(e\left(\frac{199}{1260}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 388075 }(23203,a) \;\) at \(\;a = \) e.g. 2