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Properties

Label 9787.30
Modulus $9787$
Conductor $9787$
Order $9786$
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(9787, base_ring=CyclotomicField(9786)) M = H._module chi = DirichletCharacter(H, M([5639]))
 
Copy content gp:[g,chi] = znchar(Mod(30, 9787))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9787.30");
 

Basic properties

Modulus: \(9787\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(9787\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(9786\)
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 9787.p

\(\chi_{9787}(3,\cdot)\) \(\chi_{9787}(5,\cdot)\) \(\chi_{9787}(12,\cdot)\) \(\chi_{9787}(17,\cdot)\) \(\chi_{9787}(20,\cdot)\) \(\chi_{9787}(22,\cdot)\) \(\chi_{9787}(29,\cdot)\) \(\chi_{9787}(30,\cdot)\) \(\chi_{9787}(33,\cdot)\) \(\chi_{9787}(38,\cdot)\) \(\chi_{9787}(41,\cdot)\) \(\chi_{9787}(42,\cdot)\) \(\chi_{9787}(50,\cdot)\) \(\chi_{9787}(52,\cdot)\) \(\chi_{9787}(53,\cdot)\) \(\chi_{9787}(55,\cdot)\) \(\chi_{9787}(62,\cdot)\) \(\chi_{9787}(68,\cdot)\) \(\chi_{9787}(70,\cdot)\) \(\chi_{9787}(72,\cdot)\) \(\chi_{9787}(73,\cdot)\) \(\chi_{9787}(77,\cdot)\) \(\chi_{9787}(80,\cdot)\) \(\chi_{9787}(86,\cdot)\) \(\chi_{9787}(88,\cdot)\) \(\chi_{9787}(89,\cdot)\) \(\chi_{9787}(93,\cdot)\) \(\chi_{9787}(94,\cdot)\) \(\chi_{9787}(105,\cdot)\) \(\chi_{9787}(116,\cdot)\) ...

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: $\Q(\zeta_{4893})$
Fixed field: Number field defined by a degree 9786 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{5639}{9786}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 9787 }(30, a) \) \(-1\)\(1\)\(e\left(\frac{1269}{3262}\right)\)\(e\left(\frac{5639}{9786}\right)\)\(e\left(\frac{1269}{1631}\right)\)\(e\left(\frac{3947}{9786}\right)\)\(e\left(\frac{4723}{4893}\right)\)\(e\left(\frac{961}{3262}\right)\)\(e\left(\frac{545}{3262}\right)\)\(e\left(\frac{746}{4893}\right)\)\(e\left(\frac{3877}{4893}\right)\)\(e\left(\frac{3373}{4893}\right)\)
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)
 
\( \chi_{ 9787 }(30,a) \;\) at \(\;a = \) e.g. 2