Properties

Label 319.113
Modulus 319319
Conductor 319319
Order 140140
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(319, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([112,95]))
 
pari: [g,chi] = znchar(Mod(113,319))
 

Basic properties

Modulus: 319319
Conductor: 319319
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 140140
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 319.w

χ319(3,)\chi_{319}(3,\cdot) χ319(14,)\chi_{319}(14,\cdot) χ319(15,)\chi_{319}(15,\cdot) χ319(26,)\chi_{319}(26,\cdot) χ319(27,)\chi_{319}(27,\cdot) χ319(31,)\chi_{319}(31,\cdot) χ319(37,)\chi_{319}(37,\cdot) χ319(47,)\chi_{319}(47,\cdot) χ319(48,)\chi_{319}(48,\cdot) χ319(60,)\chi_{319}(60,\cdot) χ319(69,)\chi_{319}(69,\cdot) χ319(97,)\chi_{319}(97,\cdot) χ319(102,)\chi_{319}(102,\cdot) χ319(108,)\chi_{319}(108,\cdot) χ319(113,)\chi_{319}(113,\cdot) χ319(114,)\chi_{319}(114,\cdot) χ319(119,)\chi_{319}(119,\cdot) χ319(124,)\chi_{319}(124,\cdot) χ319(126,)\chi_{319}(126,\cdot) χ319(130,)\chi_{319}(130,\cdot) χ319(135,)\chi_{319}(135,\cdot) χ319(137,)\chi_{319}(137,\cdot) χ319(147,)\chi_{319}(147,\cdot) χ319(148,)\chi_{319}(148,\cdot) χ319(159,)\chi_{319}(159,\cdot) χ319(163,)\chi_{319}(163,\cdot) χ319(185,)\chi_{319}(185,\cdot) χ319(192,)\chi_{319}(192,\cdot) χ319(201,)\chi_{319}(201,\cdot) χ319(213,)\chi_{319}(213,\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(ζ140)\Q(\zeta_{140})
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

(233,89)(233,89)(e(45),e(1928))(e\left(\frac{4}{5}\right),e\left(\frac{19}{28}\right))

First values

aa 1-111223344556677889910101212
χ319(113,a) \chi_{ 319 }(113, a) 1-111e(67140)e\left(\frac{67}{140}\right)e(111140)e\left(\frac{111}{140}\right)e(6770)e\left(\frac{67}{70}\right)e(970)e\left(\frac{9}{70}\right)e(1970)e\left(\frac{19}{70}\right)e(2635)e\left(\frac{26}{35}\right)e(61140)e\left(\frac{61}{140}\right)e(4170)e\left(\frac{41}{70}\right)e(1728)e\left(\frac{17}{28}\right)i-i
sage: chi.jacobi_sum(n)
 
χ319(113,a)   \chi_{ 319 }(113,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ319(113,))   \tau_{ a }( \chi_{ 319 }(113,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ319(113,),χ319(n,))   J(\chi_{ 319 }(113,·),\chi_{ 319 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ319(113,))  K(a,b,\chi_{ 319 }(113,·)) \; at   a,b=\; a,b = e.g. 1,2