Properties

Label 572.15
Modulus $572$
Conductor $572$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(572, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([30,12,5]))
 
pari: [g,chi] = znchar(Mod(15,572))
 

Basic properties

Modulus: \(572\)
Conductor: \(572\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 572.bs

\(\chi_{572}(15,\cdot)\) \(\chi_{572}(59,\cdot)\) \(\chi_{572}(71,\cdot)\) \(\chi_{572}(115,\cdot)\) \(\chi_{572}(119,\cdot)\) \(\chi_{572}(163,\cdot)\) \(\chi_{572}(223,\cdot)\) \(\chi_{572}(267,\cdot)\) \(\chi_{572}(279,\cdot)\) \(\chi_{572}(323,\cdot)\) \(\chi_{572}(379,\cdot)\) \(\chi_{572}(383,\cdot)\) \(\chi_{572}(423,\cdot)\) \(\chi_{572}(427,\cdot)\) \(\chi_{572}(487,\cdot)\) \(\chi_{572}(531,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((287,365,353)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{1}{12}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(15\)\(17\)\(19\)\(21\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{31}{60}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 572 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{572}(15,\cdot)) = \sum_{r\in \Z/572\Z} \chi_{572}(15,r) e\left(\frac{r}{286}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 572 }(15,·),\chi_{ 572 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{572}(15,\cdot),\chi_{572}(1,\cdot)) = \sum_{r\in \Z/572\Z} \chi_{572}(15,r) \chi_{572}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 572 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{572}(15,·)) = \sum_{r \in \Z/572\Z} \chi_{572}(15,r) e\left(\frac{1 r + 2 r^{-1}}{572}\right) = 10.6164063044+13.1101699579i \)