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Properties

Label 3496.1189
Modulus $3496$
Conductor $3496$
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(3496, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([0,33,44,24]))
 
Copy content gp:[g,chi] = znchar(Mod(1189, 3496))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3496.1189");
 

Basic properties

Modulus: \(3496\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3496\)
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 3496.co

\(\chi_{3496}(197,\cdot)\) \(\chi_{3496}(349,\cdot)\) \(\chi_{3496}(501,\cdot)\) \(\chi_{3496}(581,\cdot)\) \(\chi_{3496}(653,\cdot)\) \(\chi_{3496}(1037,\cdot)\) \(\chi_{3496}(1189,\cdot)\) \(\chi_{3496}(1645,\cdot)\) \(\chi_{3496}(1797,\cdot)\) \(\chi_{3496}(1869,\cdot)\) \(\chi_{3496}(2101,\cdot)\) \(\chi_{3496}(2325,\cdot)\) \(\chi_{3496}(2405,\cdot)\) \(\chi_{3496}(2477,\cdot)\) \(\chi_{3496}(2557,\cdot)\) \(\chi_{3496}(2709,\cdot)\) \(\chi_{3496}(2861,\cdot)\) \(\chi_{3496}(2933,\cdot)\) \(\chi_{3496}(3085,\cdot)\) \(\chi_{3496}(3389,\cdot)\)

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

\((2623,1749,553,3041)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(25\)
\( \chi_{ 3496 }(1189, a) \) \(1\)\(1\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{2}{33}\right)\)
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)
 
\( \chi_{ 3496 }(1189,a) \;\) at \(\;a = \) e.g. 2