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Properties

Label 4335.458
Modulus $4335$
Conductor $4335$
Order $68$
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(4335, base_ring=CyclotomicField(68)) M = H._module chi = DirichletCharacter(H, M([34,51,30]))
 
Copy content gp:[g,chi] = znchar(Mod(458, 4335))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4335.458");
 

Basic properties

Modulus: \(4335\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4335\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(68\)
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 4335.bx

\(\chi_{4335}(152,\cdot)\) \(\chi_{4335}(203,\cdot)\) \(\chi_{4335}(407,\cdot)\) \(\chi_{4335}(458,\cdot)\) \(\chi_{4335}(662,\cdot)\) \(\chi_{4335}(713,\cdot)\) \(\chi_{4335}(917,\cdot)\) \(\chi_{4335}(968,\cdot)\) \(\chi_{4335}(1172,\cdot)\) \(\chi_{4335}(1223,\cdot)\) \(\chi_{4335}(1427,\cdot)\) \(\chi_{4335}(1478,\cdot)\) \(\chi_{4335}(1682,\cdot)\) \(\chi_{4335}(1937,\cdot)\) \(\chi_{4335}(1988,\cdot)\) \(\chi_{4335}(2192,\cdot)\) \(\chi_{4335}(2243,\cdot)\) \(\chi_{4335}(2447,\cdot)\) \(\chi_{4335}(2498,\cdot)\) \(\chi_{4335}(2702,\cdot)\) \(\chi_{4335}(2753,\cdot)\) \(\chi_{4335}(2957,\cdot)\) \(\chi_{4335}(3008,\cdot)\) \(\chi_{4335}(3212,\cdot)\) \(\chi_{4335}(3263,\cdot)\) \(\chi_{4335}(3518,\cdot)\) \(\chi_{4335}(3722,\cdot)\) \(\chi_{4335}(3773,\cdot)\) \(\chi_{4335}(3977,\cdot)\) \(\chi_{4335}(4028,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\((2891,2602,2026)\) → \((-1,-i,e\left(\frac{15}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 4335 }(458, a) \) \(1\)\(1\)\(e\left(\frac{5}{68}\right)\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{9}{68}\right)\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{49}{68}\right)\)
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)
 
\( \chi_{ 4335 }(458,a) \;\) at \(\;a = \) e.g. 2