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Properties

Label 4425.173
Modulus $4425$
Conductor $4425$
Order $580$
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(4425, base_ring=CyclotomicField(580)) M = H._module chi = DirichletCharacter(H, M([290,319,310]))
 
Copy content gp:[g,chi] = znchar(Mod(173, 4425))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4425.173");
 

Basic properties

Modulus: \(4425\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4425\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(580\)
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 4425.bv

\(\chi_{4425}(2,\cdot)\) \(\chi_{4425}(8,\cdot)\) \(\chi_{4425}(23,\cdot)\) \(\chi_{4425}(38,\cdot)\) \(\chi_{4425}(47,\cdot)\) \(\chi_{4425}(77,\cdot)\) \(\chi_{4425}(83,\cdot)\) \(\chi_{4425}(92,\cdot)\) \(\chi_{4425}(98,\cdot)\) \(\chi_{4425}(113,\cdot)\) \(\chi_{4425}(128,\cdot)\) \(\chi_{4425}(152,\cdot)\) \(\chi_{4425}(158,\cdot)\) \(\chi_{4425}(173,\cdot)\) \(\chi_{4425}(188,\cdot)\) \(\chi_{4425}(227,\cdot)\) \(\chi_{4425}(233,\cdot)\) \(\chi_{4425}(242,\cdot)\) \(\chi_{4425}(278,\cdot)\) \(\chi_{4425}(308,\cdot)\) \(\chi_{4425}(338,\cdot)\) \(\chi_{4425}(347,\cdot)\) \(\chi_{4425}(362,\cdot)\) \(\chi_{4425}(377,\cdot)\) \(\chi_{4425}(392,\cdot)\) \(\chi_{4425}(398,\cdot)\) \(\chi_{4425}(437,\cdot)\) \(\chi_{4425}(452,\cdot)\) \(\chi_{4425}(467,\cdot)\) \(\chi_{4425}(503,\cdot)\) ...

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: $\Q(\zeta_{580})$
Fixed field: Number field defined by a degree 580 polynomial (not computed)

Values on generators

\((2951,2302,3601)\) → \((-1,e\left(\frac{11}{20}\right),e\left(\frac{31}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4425 }(173, a) \) \(-1\)\(1\)\(e\left(\frac{339}{580}\right)\)\(e\left(\frac{49}{290}\right)\)\(e\left(\frac{43}{116}\right)\)\(e\left(\frac{437}{580}\right)\)\(e\left(\frac{96}{145}\right)\)\(e\left(\frac{291}{580}\right)\)\(e\left(\frac{277}{290}\right)\)\(e\left(\frac{49}{145}\right)\)\(e\left(\frac{17}{580}\right)\)\(e\left(\frac{61}{290}\right)\)
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)
 
\( \chi_{ 4425 }(173,a) \;\) at \(\;a = \) e.g. 2