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Properties

Label 2527.374
Modulus $2527$
Conductor $2527$
Order $342$
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(2527, base_ring=CyclotomicField(342)) M = H._module chi = DirichletCharacter(H, M([57,329]))
 
Copy content gp:[g,chi] = znchar(Mod(374, 2527))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2527.374");
 

Basic properties

Modulus: \(2527\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2527\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(342\)
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 2527.cd

\(\chi_{2527}(10,\cdot)\) \(\chi_{2527}(33,\cdot)\) \(\chi_{2527}(40,\cdot)\) \(\chi_{2527}(108,\cdot)\) \(\chi_{2527}(117,\cdot)\) \(\chi_{2527}(129,\cdot)\) \(\chi_{2527}(143,\cdot)\) \(\chi_{2527}(166,\cdot)\) \(\chi_{2527}(173,\cdot)\) \(\chi_{2527}(241,\cdot)\) \(\chi_{2527}(250,\cdot)\) \(\chi_{2527}(276,\cdot)\) \(\chi_{2527}(306,\cdot)\) \(\chi_{2527}(374,\cdot)\) \(\chi_{2527}(383,\cdot)\) \(\chi_{2527}(395,\cdot)\) \(\chi_{2527}(409,\cdot)\) \(\chi_{2527}(432,\cdot)\) \(\chi_{2527}(439,\cdot)\) \(\chi_{2527}(507,\cdot)\) \(\chi_{2527}(516,\cdot)\) \(\chi_{2527}(528,\cdot)\) \(\chi_{2527}(542,\cdot)\) \(\chi_{2527}(565,\cdot)\) \(\chi_{2527}(572,\cdot)\) \(\chi_{2527}(640,\cdot)\) \(\chi_{2527}(649,\cdot)\) \(\chi_{2527}(661,\cdot)\) \(\chi_{2527}(675,\cdot)\) \(\chi_{2527}(698,\cdot)\) ...

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: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 342 polynomial (not computed)

Values on generators

\((1445,1807)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{329}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2527 }(374, a) \) \(1\)\(1\)\(e\left(\frac{101}{342}\right)\)\(e\left(\frac{151}{171}\right)\)\(e\left(\frac{101}{171}\right)\)\(e\left(\frac{5}{342}\right)\)\(e\left(\frac{61}{342}\right)\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{131}{171}\right)\)\(e\left(\frac{53}{171}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{9}{19}\right)\)
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)
 
\( \chi_{ 2527 }(374,a) \;\) at \(\;a = \) e.g. 2