Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4740, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([0,0,117,88]))
 
Copy content gp:[g,chi] = znchar(Mod(73, 4740))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4740.73");
 

Basic properties

Modulus: \(4740\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(395\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 \(\chi_{395}(73,\cdot)\)
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 4740.do

\(\chi_{4740}(13,\cdot)\) \(\chi_{4740}(73,\cdot)\) \(\chi_{4740}(253,\cdot)\) \(\chi_{4740}(277,\cdot)\) \(\chi_{4740}(313,\cdot)\) \(\chi_{4740}(397,\cdot)\) \(\chi_{4740}(493,\cdot)\) \(\chi_{4740}(637,\cdot)\) \(\chi_{4740}(913,\cdot)\) \(\chi_{4740}(973,\cdot)\) \(\chi_{4740}(997,\cdot)\) \(\chi_{4740}(1117,\cdot)\) \(\chi_{4740}(1273,\cdot)\) \(\chi_{4740}(1393,\cdot)\) \(\chi_{4740}(1453,\cdot)\) \(\chi_{4740}(1537,\cdot)\) \(\chi_{4740}(1573,\cdot)\) \(\chi_{4740}(1837,\cdot)\) \(\chi_{4740}(2017,\cdot)\) \(\chi_{4740}(2137,\cdot)\) \(\chi_{4740}(2173,\cdot)\) \(\chi_{4740}(2257,\cdot)\) \(\chi_{4740}(2293,\cdot)\) \(\chi_{4740}(2317,\cdot)\) \(\chi_{4740}(2533,\cdot)\) \(\chi_{4740}(2737,\cdot)\) \(\chi_{4740}(2797,\cdot)\) \(\chi_{4740}(2857,\cdot)\) \(\chi_{4740}(2893,\cdot)\) \(\chi_{4740}(2917,\cdot)\) ...

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: $\Q(\zeta_{156})$
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 156 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((2371,3161,1897,1741)\) → \((1,1,-i,e\left(\frac{22}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4740 }(73, a) \) \(-1\)\(1\)\(e\left(\frac{101}{156}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{67}{156}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{4}{13}\right)\)
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)
 
\( \chi_{ 4740 }(73,a) \;\) at \(\;a = \) e.g. 2