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Properties

Label 7524.2399
Modulus $7524$
Conductor $684$
Order $18$
Real no
Primitive no
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(7524, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([9,15,0,16]))
 
Copy content gp:[g,chi] = znchar(Mod(2399, 7524))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7524.2399");
 

Basic properties

Modulus: \(7524\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(684\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(18\)
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: no, induced from \(\chi_{684}(347,\cdot)\)
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 7524.ex

\(\chi_{7524}(815,\cdot)\) \(\chi_{7524}(2399,\cdot)\) \(\chi_{7524}(2531,\cdot)\) \(\chi_{7524}(5963,\cdot)\) \(\chi_{7524}(6887,\cdot)\) \(\chi_{7524}(7283,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.18.576595720370217842199656610425771655168.2

Values on generators

\((3763,6689,4105,2377)\) → \((-1,e\left(\frac{5}{6}\right),1,e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 7524 }(2399, a) \) \(1\)\(1\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(-1\)\(e\left(\frac{5}{9}\right)\)\(1\)
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)
 
\( \chi_{ 7524 }(2399,a) \;\) at \(\;a = \) e.g. 2