Properties

Label 1045.1031
Modulus $1045$
Conductor $209$
Order $90$
Real no
Primitive no
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([0,27,80]))
 
pari: [g,chi] = znchar(Mod(1031,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{209}(195,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.cj

\(\chi_{1045}(6,\cdot)\) \(\chi_{1045}(61,\cdot)\) \(\chi_{1045}(101,\cdot)\) \(\chi_{1045}(156,\cdot)\) \(\chi_{1045}(161,\cdot)\) \(\chi_{1045}(206,\cdot)\) \(\chi_{1045}(226,\cdot)\) \(\chi_{1045}(271,\cdot)\) \(\chi_{1045}(321,\cdot)\) \(\chi_{1045}(446,\cdot)\) \(\chi_{1045}(481,\cdot)\) \(\chi_{1045}(491,\cdot)\) \(\chi_{1045}(536,\cdot)\) \(\chi_{1045}(541,\cdot)\) \(\chi_{1045}(556,\cdot)\) \(\chi_{1045}(651,\cdot)\) \(\chi_{1045}(701,\cdot)\) \(\chi_{1045}(766,\cdot)\) \(\chi_{1045}(776,\cdot)\) \(\chi_{1045}(821,\cdot)\) \(\chi_{1045}(871,\cdot)\) \(\chi_{1045}(921,\cdot)\) \(\chi_{1045}(986,\cdot)\) \(\chi_{1045}(1031,\cdot)\)

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

\((837,761,496)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{8}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(1031, a) \) \(-1\)\(1\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{28}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(1031,a) \;\) at \(\;a = \) e.g. 2