Properties

Label 207.32
Modulus 207207
Conductor 207207
Order 6666
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,30]))
 
pari: [g,chi] = znchar(Mod(32,207))
 

Basic properties

Modulus: 207207
Conductor: 207207
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 6666
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 207.n

χ207(2,)\chi_{207}(2,\cdot) χ207(29,)\chi_{207}(29,\cdot) χ207(32,)\chi_{207}(32,\cdot) χ207(41,)\chi_{207}(41,\cdot) χ207(50,)\chi_{207}(50,\cdot) χ207(59,)\chi_{207}(59,\cdot) χ207(77,)\chi_{207}(77,\cdot) χ207(95,)\chi_{207}(95,\cdot) χ207(101,)\chi_{207}(101,\cdot) χ207(104,)\chi_{207}(104,\cdot) χ207(110,)\chi_{207}(110,\cdot) χ207(119,)\chi_{207}(119,\cdot) χ207(128,)\chi_{207}(128,\cdot) χ207(131,)\chi_{207}(131,\cdot) χ207(140,)\chi_{207}(140,\cdot) χ207(146,)\chi_{207}(146,\cdot) χ207(164,)\chi_{207}(164,\cdot) χ207(167,)\chi_{207}(167,\cdot) χ207(173,)\chi_{207}(173,\cdot) χ207(200,)\chi_{207}(200,\cdot)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ33)\Q(\zeta_{33})
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

(47,28)(47,28)(e(56),e(511))(e\left(\frac{5}{6}\right),e\left(\frac{5}{11}\right))

First values

aa 1-111224455778810101111131314141616
χ207(32,a) \chi_{ 207 }(32, a) 1-111e(4966)e\left(\frac{49}{66}\right)e(1633)e\left(\frac{16}{33}\right)e(4166)e\left(\frac{41}{66}\right)e(3233)e\left(\frac{32}{33}\right)e(522)e\left(\frac{5}{22}\right)e(411)e\left(\frac{4}{11}\right)e(6166)e\left(\frac{61}{66}\right)e(133)e\left(\frac{1}{33}\right)e(4766)e\left(\frac{47}{66}\right)e(3233)e\left(\frac{32}{33}\right)
sage: chi.jacobi_sum(n)
 
χ207(32,a)   \chi_{ 207 }(32,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ207(32,))   \tau_{ a }( \chi_{ 207 }(32,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ207(32,),χ207(n,))   J(\chi_{ 207 }(32,·),\chi_{ 207 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ207(32,))  K(a,b,\chi_{ 207 }(32,·)) \; at   a,b=\; a,b = e.g. 1,2