↑

Properties

Label 50286.31
Modulus $50286$
Conductor $8381$
Order $1904$
Real no
Primitive no
Minimal yes
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(50286, base_ring=CyclotomicField(1904)) M = H._module chi = DirichletCharacter(H, M([0,63,68]))
 
Copy content gp:[g,chi] = znchar(Mod(31, 50286))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50286.31");
 

Basic properties

Modulus: \(50286\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8381\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1904\)
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 \(\chi_{8381}(31,\cdot)\)
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 50286.fg

\(\chi_{50286}(31,\cdot)\) \(\chi_{50286}(163,\cdot)\) \(\chi_{50286}(211,\cdot)\) \(\chi_{50286}(235,\cdot)\) \(\chi_{50286}(301,\cdot)\) \(\chi_{50286}(379,\cdot)\) \(\chi_{50286}(385,\cdot)\) \(\chi_{50286}(541,\cdot)\) \(\chi_{50286}(583,\cdot)\) \(\chi_{50286}(619,\cdot)\) \(\chi_{50286}(649,\cdot)\) \(\chi_{50286}(793,\cdot)\) \(\chi_{50286}(889,\cdot)\) \(\chi_{50286}(913,\cdot)\) \(\chi_{50286}(925,\cdot)\) \(\chi_{50286}(949,\cdot)\) \(\chi_{50286}(1099,\cdot)\) \(\chi_{50286}(1129,\cdot)\) \(\chi_{50286}(1261,\cdot)\) \(\chi_{50286}(1297,\cdot)\) \(\chi_{50286}(1303,\cdot)\) \(\chi_{50286}(1315,\cdot)\) \(\chi_{50286}(1423,\cdot)\) \(\chi_{50286}(1489,\cdot)\)

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: $\Q(\zeta_{1904})$
Fixed field: Number field defined by a degree 1904 polynomial (not computed)

Values on generators

\((16763,34105,41617)\) → \((1,e\left(\frac{9}{272}\right),e\left(\frac{1}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 50286 }(31, a) \) \(1\)\(1\)\(e\left(\frac{691}{1904}\right)\)\(e\left(\frac{1061}{1904}\right)\)\(e\left(\frac{1245}{1904}\right)\)\(e\left(\frac{61}{476}\right)\)\(e\left(\frac{747}{952}\right)\)\(e\left(\frac{177}{1904}\right)\)\(e\left(\frac{691}{952}\right)\)\(e\left(\frac{635}{1904}\right)\)\(e\left(\frac{219}{238}\right)\)\(e\left(\frac{715}{1904}\right)\)
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)
 
\( \chi_{ 50286 }(31,a) \;\) at \(\;a = \) e.g. 2