Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,45,0,40]))
 
Copy content gp:[g,chi] = znchar(Mod(691, 2240))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2240.691");
 

Basic properties

Modulus: \(2240\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(448\)
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 \(\chi_{448}(243,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2240.fh

\(\chi_{2240}(131,\cdot)\) \(\chi_{2240}(171,\cdot)\) \(\chi_{2240}(411,\cdot)\) \(\chi_{2240}(451,\cdot)\) \(\chi_{2240}(691,\cdot)\) \(\chi_{2240}(731,\cdot)\) \(\chi_{2240}(971,\cdot)\) \(\chi_{2240}(1011,\cdot)\) \(\chi_{2240}(1251,\cdot)\) \(\chi_{2240}(1291,\cdot)\) \(\chi_{2240}(1531,\cdot)\) \(\chi_{2240}(1571,\cdot)\) \(\chi_{2240}(1811,\cdot)\) \(\chi_{2240}(1851,\cdot)\) \(\chi_{2240}(2091,\cdot)\) \(\chi_{2240}(2131,\cdot)\)

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: \(\Q(\zeta_{48})\)
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 48 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1471,1541,897,1921)\) → \((-1,e\left(\frac{15}{16}\right),1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2240 }(691, a) \) \(1\)\(1\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{5}{6}\right)\)
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)
 
\( \chi_{ 2240 }(691,a) \;\) at \(\;a = \) e.g. 2