↑

Properties

Label 1711.136
Modulus $1711$
Conductor $1711$
Order $406$
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(1711, base_ring=CyclotomicField(406)) M = H._module chi = DirichletCharacter(H, M([348,301]))
 
Copy content gp:[g,chi] = znchar(Mod(136, 1711))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1711.136");
 

Basic properties

Modulus: \(1711\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1711\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(406\)
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 1711.u

\(\chi_{1711}(23,\cdot)\) \(\chi_{1711}(24,\cdot)\) \(\chi_{1711}(52,\cdot)\) \(\chi_{1711}(54,\cdot)\) \(\chi_{1711}(65,\cdot)\) \(\chi_{1711}(82,\cdot)\) \(\chi_{1711}(83,\cdot)\) \(\chi_{1711}(103,\cdot)\) \(\chi_{1711}(111,\cdot)\) \(\chi_{1711}(132,\cdot)\) \(\chi_{1711}(136,\cdot)\) \(\chi_{1711}(141,\cdot)\) \(\chi_{1711}(152,\cdot)\) \(\chi_{1711}(161,\cdot)\) \(\chi_{1711}(165,\cdot)\) \(\chi_{1711}(168,\cdot)\) \(\chi_{1711}(170,\cdot)\) \(\chi_{1711}(190,\cdot)\) \(\chi_{1711}(210,\cdot)\) \(\chi_{1711}(219,\cdot)\) \(\chi_{1711}(227,\cdot)\) \(\chi_{1711}(268,\cdot)\) \(\chi_{1711}(286,\cdot)\) \(\chi_{1711}(297,\cdot)\) \(\chi_{1711}(306,\cdot)\) \(\chi_{1711}(313,\cdot)\) \(\chi_{1711}(326,\cdot)\) \(\chi_{1711}(335,\cdot)\) \(\chi_{1711}(339,\cdot)\) \(\chi_{1711}(342,\cdot)\) ...

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: $\Q(\zeta_{203})$
Fixed field: Number field defined by a degree 406 polynomial (not computed)

Values on generators

\((60,1654)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{43}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1711 }(136, a) \) \(-1\)\(1\)\(e\left(\frac{243}{406}\right)\)\(e\left(\frac{72}{203}\right)\)\(e\left(\frac{40}{203}\right)\)\(e\left(\frac{62}{203}\right)\)\(e\left(\frac{387}{406}\right)\)\(e\left(\frac{128}{203}\right)\)\(e\left(\frac{323}{406}\right)\)\(e\left(\frac{144}{203}\right)\)\(e\left(\frac{367}{406}\right)\)\(e\left(\frac{391}{406}\right)\)
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)
 
\( \chi_{ 1711 }(136,a) \;\) at \(\;a = \) e.g. 2