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Properties

Label 14256.3499
Modulus $14256$
Conductor $1296$
Order $108$
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(14256, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([54,27,8,0]))
 
Copy content gp:[g,chi] = znchar(Mod(3499, 14256))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("14256.3499");
 

Basic properties

Modulus: \(14256\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1296\)
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: no, induced from \(\chi_{1296}(907,\cdot)\)
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 14256.gb

\(\chi_{14256}(67,\cdot)\) \(\chi_{14256}(331,\cdot)\) \(\chi_{14256}(859,\cdot)\) \(\chi_{14256}(1123,\cdot)\) \(\chi_{14256}(1651,\cdot)\) \(\chi_{14256}(1915,\cdot)\) \(\chi_{14256}(2443,\cdot)\) \(\chi_{14256}(2707,\cdot)\) \(\chi_{14256}(3235,\cdot)\) \(\chi_{14256}(3499,\cdot)\) \(\chi_{14256}(4027,\cdot)\) \(\chi_{14256}(4291,\cdot)\) \(\chi_{14256}(4819,\cdot)\) \(\chi_{14256}(5083,\cdot)\) \(\chi_{14256}(5611,\cdot)\) \(\chi_{14256}(5875,\cdot)\) \(\chi_{14256}(6403,\cdot)\) \(\chi_{14256}(6667,\cdot)\) \(\chi_{14256}(7195,\cdot)\) \(\chi_{14256}(7459,\cdot)\) \(\chi_{14256}(7987,\cdot)\) \(\chi_{14256}(8251,\cdot)\) \(\chi_{14256}(8779,\cdot)\) \(\chi_{14256}(9043,\cdot)\) \(\chi_{14256}(9571,\cdot)\) \(\chi_{14256}(9835,\cdot)\) \(\chi_{14256}(10363,\cdot)\) \(\chi_{14256}(10627,\cdot)\) \(\chi_{14256}(11155,\cdot)\) \(\chi_{14256}(11419,\cdot)\) ...

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

\((8911,10693,5105,6481)\) → \((-1,i,e\left(\frac{2}{27}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 14256 }(3499, a) \) \(-1\)\(1\)\(e\left(\frac{103}{108}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{37}{108}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{53}{108}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{5}{36}\right)\)
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)
 
\( \chi_{ 14256 }(3499,a) \;\) at \(\;a = \) e.g. 2