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Properties

Label 2737.1053
Modulus $2737$
Conductor $2737$
Order $66$
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(2737, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([11,33,36]))
 
Copy content gp:[g,chi] = znchar(Mod(1053, 2737))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2737.1053");
 

Basic properties

Modulus: \(2737\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2737\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 2737.ch

\(\chi_{2737}(101,\cdot)\) \(\chi_{2737}(220,\cdot)\) \(\chi_{2737}(271,\cdot)\) \(\chi_{2737}(509,\cdot)\) \(\chi_{2737}(577,\cdot)\) \(\chi_{2737}(696,\cdot)\) \(\chi_{2737}(1053,\cdot)\) \(\chi_{2737}(1223,\cdot)\) \(\chi_{2737}(1291,\cdot)\) \(\chi_{2737}(1342,\cdot)\) \(\chi_{2737}(1461,\cdot)\) \(\chi_{2737}(1580,\cdot)\) \(\chi_{2737}(2005,\cdot)\) \(\chi_{2737}(2056,\cdot)\) \(\chi_{2737}(2124,\cdot)\) \(\chi_{2737}(2175,\cdot)\) \(\chi_{2737}(2243,\cdot)\) \(\chi_{2737}(2362,\cdot)\) \(\chi_{2737}(2532,\cdot)\) \(\chi_{2737}(2651,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((1956,2094,120)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2737 }(1053, a) \) \(-1\)\(1\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{8}{33}\right)\)
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)
 
\( \chi_{ 2737 }(1053,a) \;\) at \(\;a = \) e.g. 2