↑

Properties

Label 57800.1249
Modulus $57800$
Conductor $1445$
Order $136$
Real no
Primitive no
Minimal no
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(57800, base_ring=CyclotomicField(136)) M = H._module chi = DirichletCharacter(H, M([0,0,68,5]))
 
Copy content gp:[g,chi] = znchar(Mod(1249, 57800))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("57800.1249");
 

Basic properties

Modulus: \(57800\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1445\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(136\)
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 \(\chi_{1445}(1249,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 57800.hf

\(\chi_{57800}(49,\cdot)\) \(\chi_{57800}(1249,\cdot)\) \(\chi_{57800}(2049,\cdot)\) \(\chi_{57800}(3249,\cdot)\) \(\chi_{57800}(3449,\cdot)\) \(\chi_{57800}(4649,\cdot)\) \(\chi_{57800}(5449,\cdot)\) \(\chi_{57800}(6649,\cdot)\) \(\chi_{57800}(6849,\cdot)\) \(\chi_{57800}(8049,\cdot)\) \(\chi_{57800}(10049,\cdot)\) \(\chi_{57800}(11449,\cdot)\) \(\chi_{57800}(12249,\cdot)\) \(\chi_{57800}(13649,\cdot)\) \(\chi_{57800}(15649,\cdot)\) \(\chi_{57800}(16849,\cdot)\) \(\chi_{57800}(17049,\cdot)\) \(\chi_{57800}(18249,\cdot)\) \(\chi_{57800}(19049,\cdot)\) \(\chi_{57800}(20249,\cdot)\) \(\chi_{57800}(20449,\cdot)\) \(\chi_{57800}(21649,\cdot)\) \(\chi_{57800}(22449,\cdot)\) \(\chi_{57800}(23649,\cdot)\) \(\chi_{57800}(23849,\cdot)\) \(\chi_{57800}(25049,\cdot)\) \(\chi_{57800}(25849,\cdot)\) \(\chi_{57800}(27049,\cdot)\) \(\chi_{57800}(27249,\cdot)\) \(\chi_{57800}(28449,\cdot)\) ...

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: $\Q(\zeta_{136})$
Fixed field: Number field defined by a degree 136 polynomial (not computed)

Values on generators

\((43351,28901,53177,52601)\) → \((1,1,-1,e\left(\frac{5}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 57800 }(1249, a) \) \(1\)\(1\)\(e\left(\frac{73}{136}\right)\)\(e\left(\frac{27}{136}\right)\)\(e\left(\frac{5}{68}\right)\)\(e\left(\frac{115}{136}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{35}{68}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{95}{136}\right)\)\(e\left(\frac{83}{136}\right)\)\(e\left(\frac{81}{136}\right)\)
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)
 
\( \chi_{ 57800 }(1249,a) \;\) at \(\;a = \) e.g. 2