↑

Properties

Label 36667.1601
Modulus $36667$
Conductor $36667$
Order $495$
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(36667, base_ring=CyclotomicField(990)) M = H._module chi = DirichletCharacter(H, M([660,776]))
 
Copy content gp:[g,chi] = znchar(Mod(1601, 36667))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36667.1601");
 

Basic properties

Modulus: \(36667\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(36667\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(495\)
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 36667.ms

\(\chi_{36667}(10,\cdot)\) \(\chi_{36667}(100,\cdot)\) \(\chi_{36667}(121,\cdot)\) \(\chi_{36667}(248,\cdot)\) \(\chi_{36667}(380,\cdot)\) \(\chi_{36667}(528,\cdot)\) \(\chi_{36667}(1136,\cdot)\) \(\chi_{36667}(1210,\cdot)\) \(\chi_{36667}(1432,\cdot)\) \(\chi_{36667}(1527,\cdot)\) \(\chi_{36667}(1601,\cdot)\) \(\chi_{36667}(1823,\cdot)\) \(\chi_{36667}(1987,\cdot)\) \(\chi_{36667}(2246,\cdot)\) \(\chi_{36667}(2304,\cdot)\) \(\chi_{36667}(2838,\cdot)\) \(\chi_{36667}(3134,\cdot)\) \(\chi_{36667}(3229,\cdot)\) \(\chi_{36667}(3451,\cdot)\) \(\chi_{36667}(3525,\cdot)\) \(\chi_{36667}(3599,\cdot)\) \(\chi_{36667}(3800,\cdot)\) \(\chi_{36667}(4022,\cdot)\) \(\chi_{36667}(4133,\cdot)\) \(\chi_{36667}(4281,\cdot)\) \(\chi_{36667}(4302,\cdot)\) \(\chi_{36667}(4466,\cdot)\) \(\chi_{36667}(4598,\cdot)\) \(\chi_{36667}(4725,\cdot)\) \(\chi_{36667}(4799,\cdot)\) ...

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: $\Q(\zeta_{495})$
Fixed field: Number field defined by a degree 495 polynomial (not computed)

Values on generators

\((22794,32709)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{388}{495}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 36667 }(1601, a) \) \(1\)\(1\)\(e\left(\frac{391}{495}\right)\)\(e\left(\frac{164}{165}\right)\)\(e\left(\frac{287}{495}\right)\)\(e\left(\frac{157}{495}\right)\)\(e\left(\frac{388}{495}\right)\)\(e\left(\frac{2}{495}\right)\)\(e\left(\frac{61}{165}\right)\)\(e\left(\frac{163}{165}\right)\)\(e\left(\frac{53}{495}\right)\)\(e\left(\frac{283}{495}\right)\)
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)
 
\( \chi_{ 36667 }(1601,a) \;\) at \(\;a = \) e.g. 2