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Properties

Label 257725.3607
Modulus $257725$
Conductor $51545$
Order $780$
Real no
Primitive no
Minimal yes
Parity odd

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(257725, base_ring=CyclotomicField(780)) M = H._module chi = DirichletCharacter(H, M([195,205,39]))
 
Copy content gp:[g,chi] = znchar(Mod(3607, 257725))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("257725.3607");
 

Basic properties

Modulus: \(257725\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(51545\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(780\)
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 \(\chi_{51545}(3607,\cdot)\)
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 257725.did

\(\chi_{257725}(2082,\cdot)\) \(\chi_{257725}(2407,\cdot)\) \(\chi_{257725}(3607,\cdot)\) \(\chi_{257725}(3932,\cdot)\) \(\chi_{257725}(4843,\cdot)\) \(\chi_{257725}(5818,\cdot)\) \(\chi_{257725}(6368,\cdot)\) \(\chi_{257725}(7343,\cdot)\) \(\chi_{257725}(12482,\cdot)\) \(\chi_{257725}(13457,\cdot)\) \(\chi_{257725}(14007,\cdot)\) \(\chi_{257725}(14982,\cdot)\) \(\chi_{257725}(15893,\cdot)\) \(\chi_{257725}(16218,\cdot)\) \(\chi_{257725}(17418,\cdot)\) \(\chi_{257725}(17743,\cdot)\) \(\chi_{257725}(21907,\cdot)\) \(\chi_{257725}(22232,\cdot)\) \(\chi_{257725}(23432,\cdot)\) \(\chi_{257725}(23757,\cdot)\) \(\chi_{257725}(24668,\cdot)\) \(\chi_{257725}(25643,\cdot)\) \(\chi_{257725}(26193,\cdot)\) \(\chi_{257725}(27168,\cdot)\) \(\chi_{257725}(32307,\cdot)\) \(\chi_{257725}(33282,\cdot)\) \(\chi_{257725}(33832,\cdot)\) \(\chi_{257725}(34807,\cdot)\) \(\chi_{257725}(35718,\cdot)\) \(\chi_{257725}(36043,\cdot)\) ...

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((144327,193676,177451)\) → \((i,e\left(\frac{41}{156}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 257725 }(3607, a) \) \(-1\)\(1\)\(e\left(\frac{439}{780}\right)\)\(e\left(\frac{499}{780}\right)\)\(e\left(\frac{49}{390}\right)\)\(e\left(\frac{79}{390}\right)\)\(e\left(\frac{641}{780}\right)\)\(e\left(\frac{179}{260}\right)\)\(e\left(\frac{109}{390}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{199}{260}\right)\)\(e\left(\frac{5}{13}\right)\)
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)
 
\( \chi_{ 257725 }(3607,a) \;\) at \(\;a = \) e.g. 2