↑

Properties

Label 38437.3812
Modulus $38437$
Conductor $38437$
Order $204$
Real no
Primitive yes
Minimal yes
Parity odd

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(38437, base_ring=CyclotomicField(204)) M = H._module chi = DirichletCharacter(H, M([136,189,170]))
 
Copy content gp:[g,chi] = znchar(Mod(3812, 38437))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("38437.3812");
 

Basic properties

Modulus: \(38437\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(38437\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(204\)
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: yes
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 38437.ip

\(\chi_{38437}(863,\cdot)\) \(\chi_{38437}(1152,\cdot)\) \(\chi_{38437}(1262,\cdot)\) \(\chi_{38437}(1551,\cdot)\) \(\chi_{38437}(3124,\cdot)\) \(\chi_{38437}(3413,\cdot)\) \(\chi_{38437}(3523,\cdot)\) \(\chi_{38437}(3812,\cdot)\) \(\chi_{38437}(5385,\cdot)\) \(\chi_{38437}(5674,\cdot)\) \(\chi_{38437}(5784,\cdot)\) \(\chi_{38437}(6073,\cdot)\) \(\chi_{38437}(7646,\cdot)\) \(\chi_{38437}(7935,\cdot)\) \(\chi_{38437}(8045,\cdot)\) \(\chi_{38437}(8334,\cdot)\) \(\chi_{38437}(9907,\cdot)\) \(\chi_{38437}(10196,\cdot)\) \(\chi_{38437}(10306,\cdot)\) \(\chi_{38437}(10595,\cdot)\) \(\chi_{38437}(12168,\cdot)\) \(\chi_{38437}(12457,\cdot)\) \(\chi_{38437}(12567,\cdot)\) \(\chi_{38437}(12856,\cdot)\) \(\chi_{38437}(14429,\cdot)\) \(\chi_{38437}(14718,\cdot)\) \(\chi_{38437}(14828,\cdot)\) \(\chi_{38437}(15117,\cdot)\) \(\chi_{38437}(16690,\cdot)\) \(\chi_{38437}(16979,\cdot)\) ...

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: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial (not computed)

Values on generators

\((16474,30059,34392)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{63}{68}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 38437 }(3812, a) \) \(-1\)\(1\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{29}{68}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{169}{204}\right)\)\(e\left(\frac{127}{204}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{5}{204}\right)\)\(e\left(\frac{199}{204}\right)\)\(e\left(\frac{167}{204}\right)\)
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)
 
\( \chi_{ 38437 }(3812,a) \;\) at \(\;a = \) e.g. 2