Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(937024, base_ring=CyclotomicField(53240)) M = H._module chi = DirichletCharacter(H, M([0,33275,50404]))
 
Copy content gp:[g,chi] = znchar(Mod(57, 937024))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("937024.57");
 

Basic properties

Modulus: \(937024\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(468512\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(53240\)
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 \(\chi_{468512}(410005,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 937024.fx

\(\chi_{937024}(41,\cdot)\) \(\chi_{937024}(57,\cdot)\) \(\chi_{937024}(73,\cdot)\) \(\chi_{937024}(105,\cdot)\) \(\chi_{937024}(217,\cdot)\) \(\chi_{937024}(249,\cdot)\) \(\chi_{937024}(281,\cdot)\) \(\chi_{937024}(393,\cdot)\) \(\chi_{937024}(409,\cdot)\) \(\chi_{937024}(425,\cdot)\) \(\chi_{937024}(569,\cdot)\) \(\chi_{937024}(585,\cdot)\) \(\chi_{937024}(601,\cdot)\) \(\chi_{937024}(633,\cdot)\) \(\chi_{937024}(745,\cdot)\) \(\chi_{937024}(761,\cdot)\) \(\chi_{937024}(777,\cdot)\) \(\chi_{937024}(809,\cdot)\) \(\chi_{937024}(921,\cdot)\) \(\chi_{937024}(937,\cdot)\) \(\chi_{937024}(953,\cdot)\) \(\chi_{937024}(985,\cdot)\) \(\chi_{937024}(1097,\cdot)\) \(\chi_{937024}(1113,\cdot)\) \(\chi_{937024}(1161,\cdot)\) \(\chi_{937024}(1273,\cdot)\) \(\chi_{937024}(1289,\cdot)\) \(\chi_{937024}(1305,\cdot)\) \(\chi_{937024}(1337,\cdot)\) \(\chi_{937024}(1465,\cdot)\) ...

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: $\Q(\zeta_{53240})$
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 53240 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((439231,58565,688129)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{12601}{13310}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 937024 }(57, a) \) \(-1\)\(1\)\(e\left(\frac{1227}{4840}\right)\)\(e\left(\frac{24491}{53240}\right)\)\(e\left(\frac{17849}{26620}\right)\)\(e\left(\frac{1227}{2420}\right)\)\(e\left(\frac{21289}{53240}\right)\)\(e\left(\frac{9497}{13310}\right)\)\(e\left(\frac{2017}{6655}\right)\)\(e\left(\frac{40657}{53240}\right)\)\(e\left(\frac{9839}{10648}\right)\)\(e\left(\frac{3545}{5324}\right)\)
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)
 
\( \chi_{ 937024 }(57,a) \;\) at \(\;a = \) e.g. 2