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Properties

Label 5744.3
Modulus $5744$
Conductor $5744$
Order $716$
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(5744, base_ring=CyclotomicField(716)) M = H._module chi = DirichletCharacter(H, M([358,537,372]))
 
Copy content gp:[g,chi] = znchar(Mod(3, 5744))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5744.3");
 

Basic properties

Modulus: \(5744\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5744\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(716\)
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 5744.v

\(\chi_{5744}(3,\cdot)\) \(\chi_{5744}(11,\cdot)\) \(\chi_{5744}(27,\cdot)\) \(\chi_{5744}(51,\cdot)\) \(\chi_{5744}(75,\cdot)\) \(\chi_{5744}(91,\cdot)\) \(\chi_{5744}(99,\cdot)\) \(\chi_{5744}(107,\cdot)\) \(\chi_{5744}(115,\cdot)\) \(\chi_{5744}(123,\cdot)\) \(\chi_{5744}(131,\cdot)\) \(\chi_{5744}(147,\cdot)\) \(\chi_{5744}(187,\cdot)\) \(\chi_{5744}(203,\cdot)\) \(\chi_{5744}(219,\cdot)\) \(\chi_{5744}(235,\cdot)\) \(\chi_{5744}(243,\cdot)\) \(\chi_{5744}(275,\cdot)\) \(\chi_{5744}(283,\cdot)\) \(\chi_{5744}(307,\cdot)\) \(\chi_{5744}(331,\cdot)\) \(\chi_{5744}(363,\cdot)\) \(\chi_{5744}(371,\cdot)\) \(\chi_{5744}(379,\cdot)\) \(\chi_{5744}(395,\cdot)\) \(\chi_{5744}(403,\cdot)\) \(\chi_{5744}(419,\cdot)\) \(\chi_{5744}(427,\cdot)\) \(\chi_{5744}(451,\cdot)\) \(\chi_{5744}(459,\cdot)\) ...

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: $\Q(\zeta_{716})$
Fixed field: Number field defined by a degree 716 polynomial (not computed)

Values on generators

\((719,4309,2161)\) → \((-1,-i,e\left(\frac{93}{179}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 5744 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{277}{716}\right)\)\(e\left(\frac{41}{716}\right)\)\(e\left(\frac{93}{179}\right)\)\(e\left(\frac{277}{358}\right)\)\(e\left(\frac{51}{716}\right)\)\(e\left(\frac{543}{716}\right)\)\(e\left(\frac{159}{358}\right)\)\(e\left(\frac{176}{179}\right)\)\(e\left(\frac{229}{716}\right)\)\(e\left(\frac{649}{716}\right)\)
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)
 
\( \chi_{ 5744 }(3,a) \;\) at \(\;a = \) e.g. 2