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Properties

Label 1580.1303
Modulus $1580$
Conductor $1580$
Order $156$
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(1580, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([78,117,70]))
 
Copy content gp:[g,chi] = znchar(Mod(1303, 1580))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1580.1303");
 

Basic properties

Modulus: \(1580\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1580\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 1580.bt

\(\chi_{1580}(3,\cdot)\) \(\chi_{1580}(7,\cdot)\) \(\chi_{1580}(43,\cdot)\) \(\chi_{1580}(47,\cdot)\) \(\chi_{1580}(63,\cdot)\) \(\chi_{1580}(107,\cdot)\) \(\chi_{1580}(127,\cdot)\) \(\chi_{1580}(147,\cdot)\) \(\chi_{1580}(187,\cdot)\) \(\chi_{1580}(243,\cdot)\) \(\chi_{1580}(267,\cdot)\) \(\chi_{1580}(303,\cdot)\) \(\chi_{1580}(307,\cdot)\) \(\chi_{1580}(323,\cdot)\) \(\chi_{1580}(363,\cdot)\) \(\chi_{1580}(423,\cdot)\) \(\chi_{1580}(443,\cdot)\) \(\chi_{1580}(463,\cdot)\) \(\chi_{1580}(503,\cdot)\) \(\chi_{1580}(527,\cdot)\) \(\chi_{1580}(583,\cdot)\) \(\chi_{1580}(587,\cdot)\) \(\chi_{1580}(607,\cdot)\) \(\chi_{1580}(623,\cdot)\) \(\chi_{1580}(627,\cdot)\) \(\chi_{1580}(667,\cdot)\) \(\chi_{1580}(707,\cdot)\) \(\chi_{1580}(827,\cdot)\) \(\chi_{1580}(843,\cdot)\) \(\chi_{1580}(867,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((791,317,161)\) → \((-1,-i,e\left(\frac{35}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1580 }(1303, a) \) \(-1\)\(1\)\(e\left(\frac{31}{156}\right)\)\(e\left(\frac{5}{156}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{79}{156}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{31}{52}\right)\)
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)
 
\( \chi_{ 1580 }(1303,a) \;\) at \(\;a = \) e.g. 2