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Properties

Label 3737.28
Modulus $3737$
Conductor $3737$
Order $900$
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(3737, base_ring=CyclotomicField(900)) M = H._module chi = DirichletCharacter(H, M([850,99]))
 
Copy content gp:[g,chi] = znchar(Mod(28, 3737))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3737.28");
 

Basic properties

Modulus: \(3737\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3737\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(900\)
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 3737.dg

\(\chi_{3737}(3,\cdot)\) \(\chi_{3737}(28,\cdot)\) \(\chi_{3737}(40,\cdot)\) \(\chi_{3737}(67,\cdot)\) \(\chi_{3737}(99,\cdot)\) \(\chi_{3737}(104,\cdot)\) \(\chi_{3737}(136,\cdot)\) \(\chi_{3737}(139,\cdot)\) \(\chi_{3737}(141,\cdot)\) \(\chi_{3737}(151,\cdot)\) \(\chi_{3737}(152,\cdot)\) \(\chi_{3737}(173,\cdot)\) \(\chi_{3737}(176,\cdot)\) \(\chi_{3737}(210,\cdot)\) \(\chi_{3737}(213,\cdot)\) \(\chi_{3737}(250,\cdot)\) \(\chi_{3737}(252,\cdot)\) \(\chi_{3737}(263,\cdot)\) \(\chi_{3737}(300,\cdot)\) \(\chi_{3737}(321,\cdot)\) \(\chi_{3737}(337,\cdot)\) \(\chi_{3737}(354,\cdot)\) \(\chi_{3737}(358,\cdot)\) \(\chi_{3737}(411,\cdot)\) \(\chi_{3737}(432,\cdot)\) \(\chi_{3737}(465,\cdot)\) \(\chi_{3737}(502,\cdot)\) \(\chi_{3737}(539,\cdot)\) \(\chi_{3737}(543,\cdot)\) \(\chi_{3737}(558,\cdot)\) ...

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: $\Q(\zeta_{900})$
Fixed field: Number field defined by a degree 900 polynomial (not computed)

Values on generators

\((1112,2628)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{11}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3737 }(28, a) \) \(-1\)\(1\)\(e\left(\frac{49}{900}\right)\)\(e\left(\frac{131}{900}\right)\)\(e\left(\frac{49}{450}\right)\)\(e\left(\frac{163}{450}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{191}{900}\right)\)\(e\left(\frac{49}{300}\right)\)\(e\left(\frac{131}{450}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{229}{300}\right)\)
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)
 
\( \chi_{ 3737 }(28,a) \;\) at \(\;a = \) e.g. 2