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Properties

Label 1805.683
Modulus $1805$
Conductor $1805$
Order $76$
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(1805, base_ring=CyclotomicField(76)) M = H._module chi = DirichletCharacter(H, M([57,66]))
 
Copy content gp:[g,chi] = znchar(Mod(683, 1805))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1805.683");
 

Basic properties

Modulus: \(1805\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1805\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(76\)
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 1805.x

\(\chi_{1805}(18,\cdot)\) \(\chi_{1805}(37,\cdot)\) \(\chi_{1805}(113,\cdot)\) \(\chi_{1805}(132,\cdot)\) \(\chi_{1805}(208,\cdot)\) \(\chi_{1805}(227,\cdot)\) \(\chi_{1805}(303,\cdot)\) \(\chi_{1805}(322,\cdot)\) \(\chi_{1805}(398,\cdot)\) \(\chi_{1805}(417,\cdot)\) \(\chi_{1805}(493,\cdot)\) \(\chi_{1805}(512,\cdot)\) \(\chi_{1805}(588,\cdot)\) \(\chi_{1805}(607,\cdot)\) \(\chi_{1805}(683,\cdot)\) \(\chi_{1805}(702,\cdot)\) \(\chi_{1805}(778,\cdot)\) \(\chi_{1805}(797,\cdot)\) \(\chi_{1805}(873,\cdot)\) \(\chi_{1805}(892,\cdot)\) \(\chi_{1805}(968,\cdot)\) \(\chi_{1805}(987,\cdot)\) \(\chi_{1805}(1063,\cdot)\) \(\chi_{1805}(1158,\cdot)\) \(\chi_{1805}(1177,\cdot)\) \(\chi_{1805}(1253,\cdot)\) \(\chi_{1805}(1272,\cdot)\) \(\chi_{1805}(1348,\cdot)\) \(\chi_{1805}(1367,\cdot)\) \(\chi_{1805}(1462,\cdot)\) ...

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: $\Q(\zeta_{76})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 76 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((362,1446)\) → \((-i,e\left(\frac{33}{38}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(683, a) \) \(1\)\(1\)\(e\left(\frac{47}{76}\right)\)\(e\left(\frac{73}{76}\right)\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{1}{76}\right)\)\(e\left(\frac{65}{76}\right)\)\(e\left(\frac{35}{38}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{15}{76}\right)\)\(e\left(\frac{73}{76}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1805 }(683,a) \;\) at \(\;a = \) e.g. 2