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Properties

Label 312325.39044
Modulus $312325$
Conductor $312325$
Order $620$
Real no
Primitive yes
Minimal yes
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(312325, base_ring=CyclotomicField(620)) M = H._module chi = DirichletCharacter(H, M([558,465,614]))
 
Copy content gp:[g,chi] = znchar(Mod(39044, 312325))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312325.39044");
 

Basic properties

Modulus: \(312325\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(312325\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(620\)
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: yes
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 312325.bia

\(\chi_{312325}(294,\cdot)\) \(\chi_{312325}(1139,\cdot)\) \(\chi_{312325}(3034,\cdot)\) \(\chi_{312325}(4584,\cdot)\) \(\chi_{312325}(7779,\cdot)\) \(\chi_{312325}(8819,\cdot)\) \(\chi_{312325}(9329,\cdot)\) \(\chi_{312325}(9664,\cdot)\) \(\chi_{312325}(10369,\cdot)\) \(\chi_{312325}(11214,\cdot)\) \(\chi_{312325}(13109,\cdot)\) \(\chi_{312325}(14659,\cdot)\) \(\chi_{312325}(17854,\cdot)\) \(\chi_{312325}(18894,\cdot)\) \(\chi_{312325}(19404,\cdot)\) \(\chi_{312325}(19739,\cdot)\) \(\chi_{312325}(20444,\cdot)\) \(\chi_{312325}(21289,\cdot)\) \(\chi_{312325}(23184,\cdot)\) \(\chi_{312325}(24734,\cdot)\) \(\chi_{312325}(27929,\cdot)\) \(\chi_{312325}(28969,\cdot)\) \(\chi_{312325}(29479,\cdot)\) \(\chi_{312325}(29814,\cdot)\) \(\chi_{312325}(30519,\cdot)\) \(\chi_{312325}(31364,\cdot)\) \(\chi_{312325}(33259,\cdot)\) \(\chi_{312325}(34809,\cdot)\) \(\chi_{312325}(38004,\cdot)\) \(\chi_{312325}(39044,\cdot)\) ...

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: $\Q(\zeta_{620})$
Fixed field: Number field defined by a degree 620 polynomial (not computed)

Values on generators

\((87452,24026,89376)\) → \((e\left(\frac{9}{10}\right),-i,e\left(\frac{307}{310}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 312325 }(39044, a) \) \(1\)\(1\)\(e\left(\frac{379}{620}\right)\)\(e\left(\frac{9}{31}\right)\)\(e\left(\frac{69}{310}\right)\)\(e\left(\frac{559}{620}\right)\)\(e\left(\frac{237}{620}\right)\)\(e\left(\frac{517}{620}\right)\)\(e\left(\frac{18}{31}\right)\)\(e\left(\frac{65}{124}\right)\)\(e\left(\frac{159}{310}\right)\)\(e\left(\frac{154}{155}\right)\)
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)
 
\( \chi_{ 312325 }(39044,a) \;\) at \(\;a = \) e.g. 2