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Properties

Label 2572.1119
Modulus $2572$
Conductor $2572$
Order $214$
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(2572, base_ring=CyclotomicField(214)) M = H._module chi = DirichletCharacter(H, M([107,37]))
 
Copy content gp:[g,chi] = znchar(Mod(1119, 2572))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2572.1119");
 

Basic properties

Modulus: \(2572\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2572\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(214\)
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 2572.j

\(\chi_{2572}(3,\cdot)\) \(\chi_{2572}(27,\cdot)\) \(\chi_{2572}(43,\cdot)\) \(\chi_{2572}(67,\cdot)\) \(\chi_{2572}(71,\cdot)\) \(\chi_{2572}(75,\cdot)\) \(\chi_{2572}(103,\cdot)\) \(\chi_{2572}(107,\cdot)\) \(\chi_{2572}(119,\cdot)\) \(\chi_{2572}(127,\cdot)\) \(\chi_{2572}(243,\cdot)\) \(\chi_{2572}(259,\cdot)\) \(\chi_{2572}(283,\cdot)\) \(\chi_{2572}(299,\cdot)\) \(\chi_{2572}(319,\cdot)\) \(\chi_{2572}(387,\cdot)\) \(\chi_{2572}(403,\cdot)\) \(\chi_{2572}(427,\cdot)\) \(\chi_{2572}(483,\cdot)\) \(\chi_{2572}(499,\cdot)\) \(\chi_{2572}(543,\cdot)\) \(\chi_{2572}(547,\cdot)\) \(\chi_{2572}(579,\cdot)\) \(\chi_{2572}(583,\cdot)\) \(\chi_{2572}(603,\cdot)\) \(\chi_{2572}(607,\cdot)\) \(\chi_{2572}(619,\cdot)\) \(\chi_{2572}(627,\cdot)\) \(\chi_{2572}(639,\cdot)\) \(\chi_{2572}(651,\cdot)\) ...

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: $\Q(\zeta_{107})$
Fixed field: Number field defined by a degree 214 polynomial (not computed)

Values on generators

\((1287,1297)\) → \((-1,e\left(\frac{37}{214}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2572 }(1119, a) \) \(1\)\(1\)\(e\left(\frac{99}{107}\right)\)\(e\left(\frac{143}{214}\right)\)\(e\left(\frac{29}{214}\right)\)\(e\left(\frac{91}{107}\right)\)\(e\left(\frac{72}{107}\right)\)\(e\left(\frac{209}{214}\right)\)\(e\left(\frac{127}{214}\right)\)\(e\left(\frac{195}{214}\right)\)\(e\left(\frac{74}{107}\right)\)\(e\left(\frac{13}{214}\right)\)
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)
 
\( \chi_{ 2572 }(1119,a) \;\) at \(\;a = \) e.g. 2