Properties

Label 1728.571
Modulus $1728$
Conductor $1728$
Order $144$
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(1728, base_ring=CyclotomicField(144))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([72,9,16]))
 
pari: [g,chi] = znchar(Mod(571,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(1728\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(144\)
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 1728.cg

\(\chi_{1728}(43,\cdot)\) \(\chi_{1728}(67,\cdot)\) \(\chi_{1728}(115,\cdot)\) \(\chi_{1728}(139,\cdot)\) \(\chi_{1728}(187,\cdot)\) \(\chi_{1728}(211,\cdot)\) \(\chi_{1728}(259,\cdot)\) \(\chi_{1728}(283,\cdot)\) \(\chi_{1728}(331,\cdot)\) \(\chi_{1728}(355,\cdot)\) \(\chi_{1728}(403,\cdot)\) \(\chi_{1728}(427,\cdot)\) \(\chi_{1728}(475,\cdot)\) \(\chi_{1728}(499,\cdot)\) \(\chi_{1728}(547,\cdot)\) \(\chi_{1728}(571,\cdot)\) \(\chi_{1728}(619,\cdot)\) \(\chi_{1728}(643,\cdot)\) \(\chi_{1728}(691,\cdot)\) \(\chi_{1728}(715,\cdot)\) \(\chi_{1728}(763,\cdot)\) \(\chi_{1728}(787,\cdot)\) \(\chi_{1728}(835,\cdot)\) \(\chi_{1728}(859,\cdot)\) \(\chi_{1728}(907,\cdot)\) \(\chi_{1728}(931,\cdot)\) \(\chi_{1728}(979,\cdot)\) \(\chi_{1728}(1003,\cdot)\) \(\chi_{1728}(1051,\cdot)\) \(\chi_{1728}(1075,\cdot)\) ...

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: $\Q(\zeta_{144})$
Fixed field: Number field defined by a degree 144 polynomial (not computed)

Values on generators

\((703,325,1217)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1728 }(571, a) \) \(-1\)\(1\)\(e\left(\frac{89}{144}\right)\)\(e\left(\frac{65}{72}\right)\)\(e\left(\frac{37}{144}\right)\)\(e\left(\frac{119}{144}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{43}{72}\right)\)\(e\left(\frac{17}{72}\right)\)\(e\left(\frac{115}{144}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(571,a) \;\) at \(\;a = \) e.g. 2