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Properties

Label 58492.41
Modulus $58492$
Conductor $14623$
Order $58$
Real no
Primitive no
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(58492, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([0,29,9]))
 
Copy content gp:[g,chi] = znchar(Mod(41, 58492))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("58492.41");
 

Basic properties

Modulus: \(58492\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(14623\)
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: no, induced from \(\chi_{14623}(41,\cdot)\)
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 58492.et

\(\chi_{58492}(41,\cdot)\) \(\chi_{58492}(1833,\cdot)\) \(\chi_{58492}(2057,\cdot)\) \(\chi_{58492}(2085,\cdot)\) \(\chi_{58492}(2253,\cdot)\) \(\chi_{58492}(3401,\cdot)\) \(\chi_{58492}(6369,\cdot)\) \(\chi_{58492}(6789,\cdot)\) \(\chi_{58492}(9421,\cdot)\) \(\chi_{58492}(10317,\cdot)\) \(\chi_{58492}(10429,\cdot)\) \(\chi_{58492}(11101,\cdot)\) \(\chi_{58492}(12585,\cdot)\) \(\chi_{58492}(15693,\cdot)\) \(\chi_{58492}(27565,\cdot)\) \(\chi_{58492}(30421,\cdot)\) \(\chi_{58492}(36329,\cdot)\) \(\chi_{58492}(39689,\cdot)\) \(\chi_{58492}(39773,\cdot)\) \(\chi_{58492}(42041,\cdot)\) \(\chi_{58492}(43357,\cdot)\) \(\chi_{58492}(43805,\cdot)\) \(\chi_{58492}(43861,\cdot)\) \(\chi_{58492}(44197,\cdot)\) \(\chi_{58492}(46493,\cdot)\) \(\chi_{58492}(52429,\cdot)\) \(\chi_{58492}(53269,\cdot)\) \(\chi_{58492}(53857,\cdot)\)

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

\((29247,50137,54321)\) → \((1,-1,e\left(\frac{9}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 58492 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{25}{58}\right)\)\(e\left(\frac{15}{58}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{33}{58}\right)\)\(e\left(\frac{20}{29}\right)\)\(-1\)\(e\left(\frac{15}{29}\right)\)\(e\left(\frac{11}{58}\right)\)\(e\left(\frac{15}{29}\right)\)
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)
 
\( \chi_{ 58492 }(41,a) \;\) at \(\;a = \) e.g. 2