↑

Properties

Label 8640.103
Modulus $8640$
Conductor $4320$
Order $72$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

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(72)) M = H._module chi = DirichletCharacter(H, M([36,9,56,54]))
 
Copy content gp:[g,chi] = znchar(Mod(103, 8640))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8640.103");
 

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: \(4320\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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 \(\chi_{4320}(1723,\cdot)\)
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.hj

\(\chi_{8640}(103,\cdot)\) \(\chi_{8640}(247,\cdot)\) \(\chi_{8640}(583,\cdot)\) \(\chi_{8640}(727,\cdot)\) \(\chi_{8640}(1543,\cdot)\) \(\chi_{8640}(1687,\cdot)\) \(\chi_{8640}(2023,\cdot)\) \(\chi_{8640}(2167,\cdot)\) \(\chi_{8640}(2983,\cdot)\) \(\chi_{8640}(3127,\cdot)\) \(\chi_{8640}(3463,\cdot)\) \(\chi_{8640}(3607,\cdot)\) \(\chi_{8640}(4423,\cdot)\) \(\chi_{8640}(4567,\cdot)\) \(\chi_{8640}(4903,\cdot)\) \(\chi_{8640}(5047,\cdot)\) \(\chi_{8640}(5863,\cdot)\) \(\chi_{8640}(6007,\cdot)\) \(\chi_{8640}(6343,\cdot)\) \(\chi_{8640}(6487,\cdot)\) \(\chi_{8640}(7303,\cdot)\) \(\chi_{8640}(7447,\cdot)\) \(\chi_{8640}(7783,\cdot)\) \(\chi_{8640}(7927,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{7}{9}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(103, a) \) \(1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{17}{72}\right)\)\(e\left(\frac{25}{72}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{47}{72}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{35}{36}\right)\)
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)
 
\( \chi_{ 8640 }(103,a) \;\) at \(\;a = \) e.g. 2