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Properties

Label 4033.758
Modulus $4033$
Conductor $4033$
Order $108$
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(4033, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([51,22]))
 
Copy content gp:[g,chi] = znchar(Mod(758, 4033))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4033.758");
 

Basic properties

Modulus: \(4033\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4033\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(108\)
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 4033.jr

\(\chi_{4033}(61,\cdot)\) \(\chi_{4033}(106,\cdot)\) \(\chi_{4033}(246,\cdot)\) \(\chi_{4033}(301,\cdot)\) \(\chi_{4033}(387,\cdot)\) \(\chi_{4033}(647,\cdot)\) \(\chi_{4033}(758,\cdot)\) \(\chi_{4033}(799,\cdot)\) \(\chi_{4033}(846,\cdot)\) \(\chi_{4033}(1164,\cdot)\) \(\chi_{4033}(1337,\cdot)\) \(\chi_{4033}(1408,\cdot)\) \(\chi_{4033}(1626,\cdot)\) \(\chi_{4033}(1647,\cdot)\) \(\chi_{4033}(1719,\cdot)\) \(\chi_{4033}(1882,\cdot)\) \(\chi_{4033}(2050,\cdot)\) \(\chi_{4033}(2240,\cdot)\) \(\chi_{4033}(2383,\cdot)\) \(\chi_{4033}(2418,\cdot)\) \(\chi_{4033}(2609,\cdot)\) \(\chi_{4033}(2677,\cdot)\) \(\chi_{4033}(2703,\cdot)\) \(\chi_{4033}(2720,\cdot)\) \(\chi_{4033}(2862,\cdot)\) \(\chi_{4033}(2921,\cdot)\) \(\chi_{4033}(3017,\cdot)\) \(\chi_{4033}(3197,\cdot)\) \(\chi_{4033}(3572,\cdot)\) \(\chi_{4033}(3594,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1963,2295)\) → \((e\left(\frac{17}{36}\right),e\left(\frac{11}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(758, a) \) \(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{37}{108}\right)\)\(e\left(\frac{103}{108}\right)\)\(e\left(\frac{7}{27}\right)\)\(i\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{2}{27}\right)\)
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)
 
\( \chi_{ 4033 }(758,a) \;\) at \(\;a = \) e.g. 2