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Properties

Label 1580.1447
Modulus $1580$
Conductor $1580$
Order $156$
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(1580, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([78,39,92]))
 
Copy content gp:[g,chi] = znchar(Mod(1447, 1580))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1580.1447");
 

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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1580.bs

\(\chi_{1580}(83,\cdot)\) \(\chi_{1580}(123,\cdot)\) \(\chi_{1580}(163,\cdot)\) \(\chi_{1580}(167,\cdot)\) \(\chi_{1580}(183,\cdot)\) \(\chi_{1580}(203,\cdot)\) \(\chi_{1580}(207,\cdot)\) \(\chi_{1580}(263,\cdot)\) \(\chi_{1580}(287,\cdot)\) \(\chi_{1580}(327,\cdot)\) \(\chi_{1580}(347,\cdot)\) \(\chi_{1580}(367,\cdot)\) \(\chi_{1580}(427,\cdot)\) \(\chi_{1580}(467,\cdot)\) \(\chi_{1580}(483,\cdot)\) \(\chi_{1580}(487,\cdot)\) \(\chi_{1580}(523,\cdot)\) \(\chi_{1580}(547,\cdot)\) \(\chi_{1580}(603,\cdot)\) \(\chi_{1580}(643,\cdot)\) \(\chi_{1580}(663,\cdot)\) \(\chi_{1580}(683,\cdot)\) \(\chi_{1580}(727,\cdot)\) \(\chi_{1580}(743,\cdot)\) \(\chi_{1580}(747,\cdot)\) \(\chi_{1580}(783,\cdot)\) \(\chi_{1580}(787,\cdot)\) \(\chi_{1580}(803,\cdot)\) \(\chi_{1580}(863,\cdot)\) \(\chi_{1580}(967,\cdot)\) ...

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

\((791,317,161)\) → \((-1,i,e\left(\frac{23}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1580 }(1447, a) \) \(1\)\(1\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{1}{156}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{125}{156}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{27}{52}\right)\)
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)
 
\( \chi_{ 1580 }(1447,a) \;\) at \(\;a = \) e.g. 2