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Properties

Label 3519.1121
Modulus $3519$
Conductor $3519$
Order $66$
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(3519, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([55,33,21]))
 
Copy content gp:[g,chi] = znchar(Mod(1121, 3519))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3519.1121");
 

Basic properties

Modulus: \(3519\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3519\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3519.cc

\(\chi_{3519}(203,\cdot)\) \(\chi_{3519}(356,\cdot)\) \(\chi_{3519}(815,\cdot)\) \(\chi_{3519}(866,\cdot)\) \(\chi_{3519}(1019,\cdot)\) \(\chi_{3519}(1121,\cdot)\) \(\chi_{3519}(1325,\cdot)\) \(\chi_{3519}(1631,\cdot)\) \(\chi_{3519}(1937,\cdot)\) \(\chi_{3519}(2039,\cdot)\) \(\chi_{3519}(2090,\cdot)\) \(\chi_{3519}(2192,\cdot)\) \(\chi_{3519}(2498,\cdot)\) \(\chi_{3519}(2549,\cdot)\) \(\chi_{3519}(2702,\cdot)\) \(\chi_{3519}(2804,\cdot)\) \(\chi_{3519}(3110,\cdot)\) \(\chi_{3519}(3161,\cdot)\) \(\chi_{3519}(3263,\cdot)\) \(\chi_{3519}(3467,\cdot)\)

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

\((2738,2485,1684)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{7}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 3519 }(1121, a) \) \(1\)\(1\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{29}{33}\right)\)
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)
 
\( \chi_{ 3519 }(1121,a) \;\) at \(\;a = \) e.g. 2