↑

Properties

Label 1445.1084
Modulus $1445$
Conductor $1445$
Order $68$
Real no
Primitive yes
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(1445, base_ring=CyclotomicField(68)) M = H._module chi = DirichletCharacter(H, M([34,41]))
 
Copy content gp:[g,chi] = znchar(Mod(1084, 1445))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1445.1084");
 

Basic properties

Modulus: \(1445\)
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: \(68\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1445.w

\(\chi_{1445}(4,\cdot)\) \(\chi_{1445}(64,\cdot)\) \(\chi_{1445}(89,\cdot)\) \(\chi_{1445}(149,\cdot)\) \(\chi_{1445}(174,\cdot)\) \(\chi_{1445}(234,\cdot)\) \(\chi_{1445}(259,\cdot)\) \(\chi_{1445}(319,\cdot)\) \(\chi_{1445}(344,\cdot)\) \(\chi_{1445}(404,\cdot)\) \(\chi_{1445}(429,\cdot)\) \(\chi_{1445}(489,\cdot)\) \(\chi_{1445}(514,\cdot)\) \(\chi_{1445}(574,\cdot)\) \(\chi_{1445}(599,\cdot)\) \(\chi_{1445}(659,\cdot)\) \(\chi_{1445}(684,\cdot)\) \(\chi_{1445}(744,\cdot)\) \(\chi_{1445}(769,\cdot)\) \(\chi_{1445}(854,\cdot)\) \(\chi_{1445}(914,\cdot)\) \(\chi_{1445}(939,\cdot)\) \(\chi_{1445}(999,\cdot)\) \(\chi_{1445}(1024,\cdot)\) \(\chi_{1445}(1084,\cdot)\) \(\chi_{1445}(1109,\cdot)\) \(\chi_{1445}(1169,\cdot)\) \(\chi_{1445}(1254,\cdot)\) \(\chi_{1445}(1279,\cdot)\) \(\chi_{1445}(1339,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\((1157,581)\) → \((-1,e\left(\frac{41}{68}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1445 }(1084, a) \) \(1\)\(1\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{7}{68}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{11}{68}\right)\)\(e\left(\frac{65}{68}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{59}{68}\right)\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{23}{34}\right)\)
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)
 
\( \chi_{ 1445 }(1084,a) \;\) at \(\;a = \) e.g. 2