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Properties

Label 5315.464
Modulus $5315$
Conductor $5315$
Order $1062$
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(5315, base_ring=CyclotomicField(1062)) M = H._module chi = DirichletCharacter(H, M([531,425]))
 
Copy content gp:[g,chi] = znchar(Mod(464, 5315))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5315.464");
 

Basic properties

Modulus: \(5315\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5315\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1062\)
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 5315.bf

\(\chi_{5315}(24,\cdot)\) \(\chi_{5315}(29,\cdot)\) \(\chi_{5315}(39,\cdot)\) \(\chi_{5315}(54,\cdot)\) \(\chi_{5315}(69,\cdot)\) \(\chi_{5315}(79,\cdot)\) \(\chi_{5315}(84,\cdot)\) \(\chi_{5315}(114,\cdot)\) \(\chi_{5315}(164,\cdot)\) \(\chi_{5315}(179,\cdot)\) \(\chi_{5315}(189,\cdot)\) \(\chi_{5315}(214,\cdot)\) \(\chi_{5315}(219,\cdot)\) \(\chi_{5315}(249,\cdot)\) \(\chi_{5315}(264,\cdot)\) \(\chi_{5315}(294,\cdot)\) \(\chi_{5315}(319,\cdot)\) \(\chi_{5315}(359,\cdot)\) \(\chi_{5315}(369,\cdot)\) \(\chi_{5315}(384,\cdot)\) \(\chi_{5315}(389,\cdot)\) \(\chi_{5315}(399,\cdot)\) \(\chi_{5315}(419,\cdot)\) \(\chi_{5315}(424,\cdot)\) \(\chi_{5315}(429,\cdot)\) \(\chi_{5315}(449,\cdot)\) \(\chi_{5315}(464,\cdot)\) \(\chi_{5315}(479,\cdot)\) \(\chi_{5315}(519,\cdot)\) \(\chi_{5315}(534,\cdot)\) ...

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: $\Q(\zeta_{531})$
Fixed field: Number field defined by a degree 1062 polynomial (not computed)

Values on generators

\((2127,1066)\) → \((-1,e\left(\frac{425}{1062}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 5315 }(464, a) \) \(-1\)\(1\)\(e\left(\frac{953}{1062}\right)\)\(e\left(\frac{478}{531}\right)\)\(e\left(\frac{422}{531}\right)\)\(e\left(\frac{847}{1062}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{245}{354}\right)\)\(e\left(\frac{425}{531}\right)\)\(e\left(\frac{178}{531}\right)\)\(e\left(\frac{41}{59}\right)\)\(e\left(\frac{203}{354}\right)\)
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)
 
\( \chi_{ 5315 }(464,a) \;\) at \(\;a = \) e.g. 2