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Properties

Label 2301.1130
Modulus $2301$
Conductor $2301$
Order $58$
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(2301, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([29,29,42]))
 
Copy content gp:[g,chi] = znchar(Mod(1130, 2301))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2301.1130");
 

Basic properties

Modulus: \(2301\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2301\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
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 2301.z

\(\chi_{2301}(116,\cdot)\) \(\chi_{2301}(194,\cdot)\) \(\chi_{2301}(272,\cdot)\) \(\chi_{2301}(311,\cdot)\) \(\chi_{2301}(389,\cdot)\) \(\chi_{2301}(428,\cdot)\) \(\chi_{2301}(584,\cdot)\) \(\chi_{2301}(779,\cdot)\) \(\chi_{2301}(818,\cdot)\) \(\chi_{2301}(1052,\cdot)\) \(\chi_{2301}(1091,\cdot)\) \(\chi_{2301}(1130,\cdot)\) \(\chi_{2301}(1169,\cdot)\) \(\chi_{2301}(1208,\cdot)\) \(\chi_{2301}(1325,\cdot)\) \(\chi_{2301}(1364,\cdot)\) \(\chi_{2301}(1403,\cdot)\) \(\chi_{2301}(1442,\cdot)\) \(\chi_{2301}(1520,\cdot)\) \(\chi_{2301}(1559,\cdot)\) \(\chi_{2301}(1598,\cdot)\) \(\chi_{2301}(1715,\cdot)\) \(\chi_{2301}(1832,\cdot)\) \(\chi_{2301}(1910,\cdot)\) \(\chi_{2301}(1988,\cdot)\) \(\chi_{2301}(2027,\cdot)\) \(\chi_{2301}(2144,\cdot)\) \(\chi_{2301}(2261,\cdot)\)

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: $\Q(\zeta_{29})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 58 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1535,886,2185)\) → \((-1,-1,e\left(\frac{21}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 2301 }(1130, a) \) \(-1\)\(1\)\(e\left(\frac{21}{29}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{31}{58}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{15}{58}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{27}{58}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 2301 }(1130,a) \;\) at \(\;a = \) e.g. 2