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Properties

Label 8640.3851
Modulus $8640$
Conductor $576$
Order $48$
Real no
Primitive no
Minimal no
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(8640, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,15,40,0]))
 
Copy content gp:[g,chi] = znchar(Mod(3851, 8640))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8640.3851");
 

Basic properties

Modulus: \(8640\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(576\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(48\)
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 \(\chi_{576}(203,\cdot)\)
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: no
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 8640.gt

\(\chi_{8640}(251,\cdot)\) \(\chi_{8640}(611,\cdot)\) \(\chi_{8640}(1331,\cdot)\) \(\chi_{8640}(1691,\cdot)\) \(\chi_{8640}(2411,\cdot)\) \(\chi_{8640}(2771,\cdot)\) \(\chi_{8640}(3491,\cdot)\) \(\chi_{8640}(3851,\cdot)\) \(\chi_{8640}(4571,\cdot)\) \(\chi_{8640}(4931,\cdot)\) \(\chi_{8640}(5651,\cdot)\) \(\chi_{8640}(6011,\cdot)\) \(\chi_{8640}(6731,\cdot)\) \(\chi_{8640}(7091,\cdot)\) \(\chi_{8640}(7811,\cdot)\) \(\chi_{8640}(8171,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{5}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(3851, a) \) \(1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(i\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{13}{24}\right)\)
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)
 
\( \chi_{ 8640 }(3851,a) \;\) at \(\;a = \) e.g. 2