Properties

Label 1045.71
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([0,36,35]))
 
pari: [g,chi] = znchar(Mod(71,1045))
 

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 \(\chi_{209}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.co

\(\chi_{1045}(71,\cdot)\) \(\chi_{1045}(86,\cdot)\) \(\chi_{1045}(91,\cdot)\) \(\chi_{1045}(136,\cdot)\) \(\chi_{1045}(146,\cdot)\) \(\chi_{1045}(181,\cdot)\) \(\chi_{1045}(306,\cdot)\) \(\chi_{1045}(356,\cdot)\) \(\chi_{1045}(401,\cdot)\) \(\chi_{1045}(421,\cdot)\) \(\chi_{1045}(466,\cdot)\) \(\chi_{1045}(471,\cdot)\) \(\chi_{1045}(526,\cdot)\) \(\chi_{1045}(566,\cdot)\) \(\chi_{1045}(621,\cdot)\) \(\chi_{1045}(641,\cdot)\) \(\chi_{1045}(686,\cdot)\) \(\chi_{1045}(751,\cdot)\) \(\chi_{1045}(801,\cdot)\) \(\chi_{1045}(851,\cdot)\) \(\chi_{1045}(896,\cdot)\) \(\chi_{1045}(906,\cdot)\) \(\chi_{1045}(971,\cdot)\) \(\chi_{1045}(1021,\cdot)\)

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

\((837,761,496)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{7}{18}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{83}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(71,a) \;\) at \(\;a = \) e.g. 2