Properties

Label 1045.14
Modulus $1045$
Conductor $1045$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1045, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([45,72,35]))
 
pari: [g,chi] = znchar(Mod(14,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.cl

\(\chi_{1045}(14,\cdot)\) \(\chi_{1045}(59,\cdot)\) \(\chi_{1045}(124,\cdot)\) \(\chi_{1045}(174,\cdot)\) \(\chi_{1045}(224,\cdot)\) \(\chi_{1045}(269,\cdot)\) \(\chi_{1045}(279,\cdot)\) \(\chi_{1045}(344,\cdot)\) \(\chi_{1045}(394,\cdot)\) \(\chi_{1045}(489,\cdot)\) \(\chi_{1045}(504,\cdot)\) \(\chi_{1045}(509,\cdot)\) \(\chi_{1045}(554,\cdot)\) \(\chi_{1045}(564,\cdot)\) \(\chi_{1045}(599,\cdot)\) \(\chi_{1045}(724,\cdot)\) \(\chi_{1045}(774,\cdot)\) \(\chi_{1045}(819,\cdot)\) \(\chi_{1045}(839,\cdot)\) \(\chi_{1045}(884,\cdot)\) \(\chi_{1045}(889,\cdot)\) \(\chi_{1045}(944,\cdot)\) \(\chi_{1045}(984,\cdot)\) \(\chi_{1045}(1039,\cdot)\)

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

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((837,761,496)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{18}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{11}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(14,a) \;\) at \(\;a = \) e.g. 2