Properties

Label 567.191
Modulus 567567
Conductor 567567
Order 5454
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([37,18]))
 
Copy content pari:[g,chi] = znchar(Mod(191,567))
 

Basic properties

Modulus: 567567
Conductor: 567567
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 5454
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 567.bq

χ567(2,)\chi_{567}(2,\cdot) χ567(32,)\chi_{567}(32,\cdot) χ567(65,)\chi_{567}(65,\cdot) χ567(95,)\chi_{567}(95,\cdot) χ567(128,)\chi_{567}(128,\cdot) χ567(158,)\chi_{567}(158,\cdot) χ567(191,)\chi_{567}(191,\cdot) χ567(221,)\chi_{567}(221,\cdot) χ567(254,)\chi_{567}(254,\cdot) χ567(284,)\chi_{567}(284,\cdot) χ567(317,)\chi_{567}(317,\cdot) χ567(347,)\chi_{567}(347,\cdot) χ567(380,)\chi_{567}(380,\cdot) χ567(410,)\chi_{567}(410,\cdot) χ567(443,)\chi_{567}(443,\cdot) χ567(473,)\chi_{567}(473,\cdot) χ567(506,)\chi_{567}(506,\cdot) χ567(536,)\chi_{567}(536,\cdot)

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

Related number fields

Field of values: Q(ζ27)\Q(\zeta_{27})
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

(407,325)(407,325)(e(3754),e(13))(e\left(\frac{37}{54}\right),e\left(\frac{1}{3}\right))

First values

aa 1-11122445588101011111313161617171919
χ567(191,a) \chi_{ 567 }(191, a) 1-111e(1954)e\left(\frac{19}{54}\right)e(1927)e\left(\frac{19}{27}\right)e(2354)e\left(\frac{23}{54}\right)e(118)e\left(\frac{1}{18}\right)e(79)e\left(\frac{7}{9}\right)e(1354)e\left(\frac{13}{54}\right)e(1327)e\left(\frac{13}{27}\right)e(1127)e\left(\frac{11}{27}\right)e(1718)e\left(\frac{17}{18}\right)e(59)e\left(\frac{5}{9}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ567(191,a)   \chi_{ 567 }(191,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ567(191,))   \tau_{ a }( \chi_{ 567 }(191,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ567(191,),χ567(n,))   J(\chi_{ 567 }(191,·),\chi_{ 567 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ567(191,))  K(a,b,\chi_{ 567 }(191,·)) \; at   a,b=\; a,b = e.g. 1,2