Properties

Label 2736.143
Modulus $2736$
Conductor $228$
Order $18$
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(2736, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,0,9,17]))
 
pari: [g,chi] = znchar(Mod(143,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(228\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{228}(143,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.ft

\(\chi_{2736}(143,\cdot)\) \(\chi_{2736}(287,\cdot)\) \(\chi_{2736}(431,\cdot)\) \(\chi_{2736}(1295,\cdot)\) \(\chi_{2736}(1439,\cdot)\) \(\chi_{2736}(2447,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.0.28277592430157040563214702870528.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,1,-1,e\left(\frac{17}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(143,a) \;\) at \(\;a = \) e.g. 2