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Properties

Label 1291.76
Modulus $1291$
Conductor $1291$
Order $645$
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(1291, base_ring=CyclotomicField(1290)) M = H._module chi = DirichletCharacter(H, M([52]))
 
Copy content gp:[g,chi] = znchar(Mod(76, 1291))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1291.76");
 

Basic properties

Modulus: \(1291\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1291\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(645\)
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 1291.o

\(\chi_{1291}(4,\cdot)\) \(\chi_{1291}(7,\cdot)\) \(\chi_{1291}(9,\cdot)\) \(\chi_{1291}(13,\cdot)\) \(\chi_{1291}(16,\cdot)\) \(\chi_{1291}(17,\cdot)\) \(\chi_{1291}(20,\cdot)\) \(\chi_{1291}(33,\cdot)\) \(\chi_{1291}(35,\cdot)\) \(\chi_{1291}(37,\cdot)\) \(\chi_{1291}(41,\cdot)\) \(\chi_{1291}(45,\cdot)\) \(\chi_{1291}(46,\cdot)\) \(\chi_{1291}(49,\cdot)\) \(\chi_{1291}(54,\cdot)\) \(\chi_{1291}(58,\cdot)\) \(\chi_{1291}(59,\cdot)\) \(\chi_{1291}(62,\cdot)\) \(\chi_{1291}(63,\cdot)\) \(\chi_{1291}(73,\cdot)\) \(\chi_{1291}(76,\cdot)\) \(\chi_{1291}(78,\cdot)\) \(\chi_{1291}(79,\cdot)\) \(\chi_{1291}(80,\cdot)\) \(\chi_{1291}(81,\cdot)\) \(\chi_{1291}(86,\cdot)\) \(\chi_{1291}(88,\cdot)\) \(\chi_{1291}(89,\cdot)\) \(\chi_{1291}(95,\cdot)\) \(\chi_{1291}(96,\cdot)\) ...

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: $\Q(\zeta_{645})$
Fixed field: Number field defined by a degree 645 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{26}{645}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1291 }(76, a) \) \(1\)\(1\)\(e\left(\frac{26}{645}\right)\)\(e\left(\frac{67}{645}\right)\)\(e\left(\frac{52}{645}\right)\)\(e\left(\frac{124}{215}\right)\)\(e\left(\frac{31}{215}\right)\)\(e\left(\frac{287}{645}\right)\)\(e\left(\frac{26}{215}\right)\)\(e\left(\frac{134}{645}\right)\)\(e\left(\frac{398}{645}\right)\)\(e\left(\frac{116}{129}\right)\)
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)
 
\( \chi_{ 1291 }(76,a) \;\) at \(\;a = \) e.g. 2