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Properties

Label 7683.488
Modulus $7683$
Conductor $7683$
Order $588$
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(7683, base_ring=CyclotomicField(588)) M = H._module chi = DirichletCharacter(H, M([294,539,201]))
 
Copy content gp:[g,chi] = znchar(Mod(488, 7683))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7683.488");
 

Basic properties

Modulus: \(7683\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7683\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(588\)
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 7683.ei

\(\chi_{7683}(2,\cdot)\) \(\chi_{7683}(32,\cdot)\) \(\chi_{7683}(50,\cdot)\) \(\chi_{7683}(80,\cdot)\) \(\chi_{7683}(89,\cdot)\) \(\chi_{7683}(119,\cdot)\) \(\chi_{7683}(215,\cdot)\) \(\chi_{7683}(254,\cdot)\) \(\chi_{7683}(305,\cdot)\) \(\chi_{7683}(344,\cdot)\) \(\chi_{7683}(362,\cdot)\) \(\chi_{7683}(461,\cdot)\) \(\chi_{7683}(488,\cdot)\) \(\chi_{7683}(500,\cdot)\) \(\chi_{7683}(509,\cdot)\) \(\chi_{7683}(518,\cdot)\) \(\chi_{7683}(539,\cdot)\) \(\chi_{7683}(635,\cdot)\) \(\chi_{7683}(665,\cdot)\) \(\chi_{7683}(743,\cdot)\) \(\chi_{7683}(800,\cdot)\) \(\chi_{7683}(860,\cdot)\) \(\chi_{7683}(890,\cdot)\) \(\chi_{7683}(899,\cdot)\) \(\chi_{7683}(968,\cdot)\) \(\chi_{7683}(977,\cdot)\) \(\chi_{7683}(1016,\cdot)\) \(\chi_{7683}(1064,\cdot)\) \(\chi_{7683}(1151,\cdot)\) \(\chi_{7683}(1190,\cdot)\) ...

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: $\Q(\zeta_{588})$
Fixed field: Number field defined by a degree 588 polynomial (not computed)

Values on generators

\((5123,6502,3745)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{67}{196}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 7683 }(488, a) \) \(-1\)\(1\)\(e\left(\frac{223}{294}\right)\)\(e\left(\frac{76}{147}\right)\)\(e\left(\frac{17}{98}\right)\)\(e\left(\frac{583}{588}\right)\)\(e\left(\frac{27}{98}\right)\)\(e\left(\frac{137}{147}\right)\)\(e\left(\frac{122}{147}\right)\)\(-i\)\(e\left(\frac{5}{147}\right)\)\(e\left(\frac{403}{588}\right)\)
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)
 
\( \chi_{ 7683 }(488,a) \;\) at \(\;a = \) e.g. 2