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Properties

Label 5780.623
Modulus $5780$
Conductor $5780$
Order $272$
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(5780, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([136,204,231]))
 
Copy content gp:[g,chi] = znchar(Mod(623, 5780))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5780.623");
 

Basic properties

Modulus: \(5780\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5780\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(272\)
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 5780.cm

\(\chi_{5780}(23,\cdot)\) \(\chi_{5780}(107,\cdot)\) \(\chi_{5780}(143,\cdot)\) \(\chi_{5780}(163,\cdot)\) \(\chi_{5780}(167,\cdot)\) \(\chi_{5780}(207,\cdot)\) \(\chi_{5780}(267,\cdot)\) \(\chi_{5780}(283,\cdot)\) \(\chi_{5780}(363,\cdot)\) \(\chi_{5780}(483,\cdot)\) \(\chi_{5780}(507,\cdot)\) \(\chi_{5780}(547,\cdot)\) \(\chi_{5780}(607,\cdot)\) \(\chi_{5780}(623,\cdot)\) \(\chi_{5780}(703,\cdot)\) \(\chi_{5780}(787,\cdot)\) \(\chi_{5780}(823,\cdot)\) \(\chi_{5780}(843,\cdot)\) \(\chi_{5780}(847,\cdot)\) \(\chi_{5780}(887,\cdot)\) \(\chi_{5780}(947,\cdot)\) \(\chi_{5780}(963,\cdot)\) \(\chi_{5780}(1043,\cdot)\) \(\chi_{5780}(1127,\cdot)\) \(\chi_{5780}(1163,\cdot)\) \(\chi_{5780}(1183,\cdot)\) \(\chi_{5780}(1187,\cdot)\) \(\chi_{5780}(1227,\cdot)\) \(\chi_{5780}(1303,\cdot)\) \(\chi_{5780}(1383,\cdot)\) ...

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: $\Q(\zeta_{272})$
Fixed field: Number field defined by a degree 272 polynomial (not computed)

Values on generators

\((2891,1157,581)\) → \((-1,-i,e\left(\frac{231}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 5780 }(623, a) \) \(-1\)\(1\)\(e\left(\frac{163}{272}\right)\)\(e\left(\frac{241}{272}\right)\)\(e\left(\frac{27}{136}\right)\)\(e\left(\frac{9}{272}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{121}{136}\right)\)\(e\left(\frac{33}{68}\right)\)\(e\left(\frac{37}{272}\right)\)\(e\left(\frac{217}{272}\right)\)\(e\left(\frac{179}{272}\right)\)
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)
 
\( \chi_{ 5780 }(623,a) \;\) at \(\;a = \) e.g. 2