Properties

Label 1045.507
Modulus $1045$
Conductor $95$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1045, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([9,0,10]))
 
Copy content pari:[g,chi] = znchar(Mod(507,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(95\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(36\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(32,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1045.cc

\(\chi_{1045}(67,\cdot)\) \(\chi_{1045}(78,\cdot)\) \(\chi_{1045}(243,\cdot)\) \(\chi_{1045}(287,\cdot)\) \(\chi_{1045}(298,\cdot)\) \(\chi_{1045}(452,\cdot)\) \(\chi_{1045}(507,\cdot)\) \(\chi_{1045}(573,\cdot)\) \(\chi_{1045}(782,\cdot)\) \(\chi_{1045}(793,\cdot)\) \(\chi_{1045}(903,\cdot)\) \(\chi_{1045}(1002,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{95})^+\)

Values on generators

\((837,761,496)\) → \((i,1,e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(507, a) \) \(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{4}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(507,a) \;\) at \(\;a = \) e.g. 2