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Properties

Label 1007.309
Modulus $1007$
Conductor $1007$
Order $117$
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(1007, base_ring=CyclotomicField(234)) M = H._module chi = DirichletCharacter(H, M([208,36]))
 
Copy content gp:[g,chi] = znchar(Mod(309, 1007))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1007.309");
 

Basic properties

Modulus: \(1007\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1007\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(117\)
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 1007.bc

\(\chi_{1007}(16,\cdot)\) \(\chi_{1007}(24,\cdot)\) \(\chi_{1007}(28,\cdot)\) \(\chi_{1007}(36,\cdot)\) \(\chi_{1007}(42,\cdot)\) \(\chi_{1007}(44,\cdot)\) \(\chi_{1007}(47,\cdot)\) \(\chi_{1007}(63,\cdot)\) \(\chi_{1007}(66,\cdot)\) \(\chi_{1007}(81,\cdot)\) \(\chi_{1007}(99,\cdot)\) \(\chi_{1007}(100,\cdot)\) \(\chi_{1007}(119,\cdot)\) \(\chi_{1007}(130,\cdot)\) \(\chi_{1007}(142,\cdot)\) \(\chi_{1007}(150,\cdot)\) \(\chi_{1007}(169,\cdot)\) \(\chi_{1007}(175,\cdot)\) \(\chi_{1007}(187,\cdot)\) \(\chi_{1007}(195,\cdot)\) \(\chi_{1007}(206,\cdot)\) \(\chi_{1007}(225,\cdot)\) \(\chi_{1007}(256,\cdot)\) \(\chi_{1007}(275,\cdot)\) \(\chi_{1007}(289,\cdot)\) \(\chi_{1007}(301,\cdot)\) \(\chi_{1007}(309,\cdot)\) \(\chi_{1007}(328,\cdot)\) \(\chi_{1007}(346,\cdot)\) \(\chi_{1007}(365,\cdot)\) ...

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: $\Q(\zeta_{117})$
Fixed field: Number field defined by a degree 117 polynomial (not computed)

Values on generators

\((743,267)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{2}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1007 }(309, a) \) \(1\)\(1\)\(e\left(\frac{5}{117}\right)\)\(e\left(\frac{20}{117}\right)\)\(e\left(\frac{10}{117}\right)\)\(e\left(\frac{53}{117}\right)\)\(e\left(\frac{25}{117}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{40}{117}\right)\)\(e\left(\frac{58}{117}\right)\)\(e\left(\frac{23}{39}\right)\)
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)
 
\( \chi_{ 1007 }(309,a) \;\) at \(\;a = \) e.g. 2