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Properties

Label 2523.419
Modulus $2523$
Conductor $2523$
Order $406$
Real no
Primitive yes
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(2523, base_ring=CyclotomicField(406)) M = H._module chi = DirichletCharacter(H, M([203,65]))
 
Copy content gp:[g,chi] = znchar(Mod(419, 2523))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2523.419");
 

Basic properties

Modulus: \(2523\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2523\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(406\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2523.v

\(\chi_{2523}(5,\cdot)\) \(\chi_{2523}(35,\cdot)\) \(\chi_{2523}(38,\cdot)\) \(\chi_{2523}(62,\cdot)\) \(\chi_{2523}(71,\cdot)\) \(\chi_{2523}(80,\cdot)\) \(\chi_{2523}(92,\cdot)\) \(\chi_{2523}(122,\cdot)\) \(\chi_{2523}(125,\cdot)\) \(\chi_{2523}(149,\cdot)\) \(\chi_{2523}(158,\cdot)\) \(\chi_{2523}(167,\cdot)\) \(\chi_{2523}(179,\cdot)\) \(\chi_{2523}(209,\cdot)\) \(\chi_{2523}(212,\cdot)\) \(\chi_{2523}(245,\cdot)\) \(\chi_{2523}(254,\cdot)\) \(\chi_{2523}(266,\cdot)\) \(\chi_{2523}(296,\cdot)\) \(\chi_{2523}(299,\cdot)\) \(\chi_{2523}(323,\cdot)\) \(\chi_{2523}(332,\cdot)\) \(\chi_{2523}(341,\cdot)\) \(\chi_{2523}(353,\cdot)\) \(\chi_{2523}(383,\cdot)\) \(\chi_{2523}(386,\cdot)\) \(\chi_{2523}(410,\cdot)\) \(\chi_{2523}(419,\cdot)\) \(\chi_{2523}(428,\cdot)\) \(\chi_{2523}(440,\cdot)\) ...

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: $\Q(\zeta_{203})$
Fixed field: Number field defined by a degree 406 polynomial (not computed)

Values on generators

\((842,1684)\) → \((-1,e\left(\frac{65}{406}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2523 }(419, a) \) \(-1\)\(1\)\(e\left(\frac{134}{203}\right)\)\(e\left(\frac{65}{203}\right)\)\(e\left(\frac{345}{406}\right)\)\(e\left(\frac{40}{203}\right)\)\(e\left(\frac{199}{203}\right)\)\(e\left(\frac{207}{406}\right)\)\(e\left(\frac{95}{203}\right)\)\(e\left(\frac{200}{203}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{130}{203}\right)\)
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)
 
\( \chi_{ 2523 }(419,a) \;\) at \(\;a = \) e.g. 2