Properties

Label 1140.143
Modulus $1140$
Conductor $1140$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1140, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,18,27,34]))
 
pari: [g,chi] = znchar(Mod(143,1140))
 

Basic properties

Modulus: \(1140\)
Conductor: \(1140\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1140.cm

\(\chi_{1140}(143,\cdot)\) \(\chi_{1140}(167,\cdot)\) \(\chi_{1140}(203,\cdot)\) \(\chi_{1140}(287,\cdot)\) \(\chi_{1140}(383,\cdot)\) \(\chi_{1140}(527,\cdot)\) \(\chi_{1140}(623,\cdot)\) \(\chi_{1140}(743,\cdot)\) \(\chi_{1140}(827,\cdot)\) \(\chi_{1140}(887,\cdot)\) \(\chi_{1140}(983,\cdot)\) \(\chi_{1140}(1067,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.5957649898872336469465126830720741129625268010624803328000000000000000000000000000.1

Values on generators

\((571,761,457,781)\) → \((-1,-1,-i,e\left(\frac{17}{18}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{31}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1140 }(143,a) \;\) at \(\;a = \) e.g. 2