Properties

Label 2156.811
Modulus 21562156
Conductor 21562156
Order 7070
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2156, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,5,21]))
 
Copy content pari:[g,chi] = znchar(Mod(811,2156))
 

Basic properties

Modulus: 21562156
Conductor: 21562156
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 7070
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 2156.cc

χ2156(83,)\chi_{2156}(83,\cdot) χ2156(139,)\chi_{2156}(139,\cdot) χ2156(167,)\chi_{2156}(167,\cdot) χ2156(447,)\chi_{2156}(447,\cdot) χ2156(475,)\chi_{2156}(475,\cdot) χ2156(503,)\chi_{2156}(503,\cdot) χ2156(699,)\chi_{2156}(699,\cdot) χ2156(755,)\chi_{2156}(755,\cdot) χ2156(811,)\chi_{2156}(811,\cdot) χ2156(1007,)\chi_{2156}(1007,\cdot) χ2156(1063,)\chi_{2156}(1063,\cdot) χ2156(1091,)\chi_{2156}(1091,\cdot) χ2156(1119,)\chi_{2156}(1119,\cdot) χ2156(1315,)\chi_{2156}(1315,\cdot) χ2156(1399,)\chi_{2156}(1399,\cdot) χ2156(1427,)\chi_{2156}(1427,\cdot) χ2156(1623,)\chi_{2156}(1623,\cdot) χ2156(1679,)\chi_{2156}(1679,\cdot) χ2156(1707,)\chi_{2156}(1707,\cdot) χ2156(1735,)\chi_{2156}(1735,\cdot) χ2156(1931,)\chi_{2156}(1931,\cdot) χ2156(1987,)\chi_{2156}(1987,\cdot) χ2156(2015,)\chi_{2156}(2015,\cdot) χ2156(2043,)\chi_{2156}(2043,\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(ζ35)\Q(\zeta_{35})
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

(1079,1277,981)(1079,1277,981)(1,e(114),e(310))(-1,e\left(\frac{1}{14}\right),e\left(\frac{3}{10}\right))

First values

aa 1-1113355991313151517171919232325252727
χ2156(811,a) \chi_{ 2156 }(811, a) 1-111e(3435)e\left(\frac{34}{35}\right)e(1970)e\left(\frac{19}{70}\right)e(3335)e\left(\frac{33}{35}\right)e(2335)e\left(\frac{23}{35}\right)e(1770)e\left(\frac{17}{70}\right)e(1735)e\left(\frac{17}{35}\right)e(910)e\left(\frac{9}{10}\right)e(314)e\left(\frac{3}{14}\right)e(1935)e\left(\frac{19}{35}\right)e(3235)e\left(\frac{32}{35}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ2156(811,a)   \chi_{ 2156 }(811,a) \; at   a=\;a = e.g. 2