Properties

Label 1037.333
Modulus 10371037
Conductor 10371037
Order 8080
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1037, base_ring=CyclotomicField(80)) M = H._module chi = DirichletCharacter(H, M([15,68]))
 
Copy content pari:[g,chi] = znchar(Mod(333,1037))
 

Basic properties

Modulus: 10371037
Conductor: 10371037
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 8080
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: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1037.ck

χ1037(28,)\chi_{1037}(28,\cdot) χ1037(114,)\chi_{1037}(114,\cdot) χ1037(130,)\chi_{1037}(130,\cdot) χ1037(159,)\chi_{1037}(159,\cdot) χ1037(160,)\chi_{1037}(160,\cdot) χ1037(175,)\chi_{1037}(175,\cdot) χ1037(207,)\chi_{1037}(207,\cdot) χ1037(211,)\chi_{1037}(211,\cdot) χ1037(216,)\chi_{1037}(216,\cdot) χ1037(267,)\chi_{1037}(267,\cdot) χ1037(277,)\chi_{1037}(277,\cdot) χ1037(282,)\chi_{1037}(282,\cdot) χ1037(313,)\chi_{1037}(313,\cdot) χ1037(328,)\chi_{1037}(328,\cdot) χ1037(333,)\chi_{1037}(333,\cdot) χ1037(435,)\chi_{1037}(435,\cdot) χ1037(465,)\chi_{1037}(465,\cdot) χ1037(516,)\chi_{1037}(516,\cdot) χ1037(541,)\chi_{1037}(541,\cdot) χ1037(573,)\chi_{1037}(573,\cdot) χ1037(618,)\chi_{1037}(618,\cdot) χ1037(634,)\chi_{1037}(634,\cdot) χ1037(643,)\chi_{1037}(643,\cdot) χ1037(694,)\chi_{1037}(694,\cdot) χ1037(708,)\chi_{1037}(708,\cdot) χ1037(785,)\chi_{1037}(785,\cdot) χ1037(887,)\chi_{1037}(887,\cdot) χ1037(891,)\chi_{1037}(891,\cdot) χ1037(938,)\chi_{1037}(938,\cdot) χ1037(1000,)\chi_{1037}(1000,\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(ζ80)\Q(\zeta_{80})
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

(428,307)(428,307)(e(316),e(1720))(e\left(\frac{3}{16}\right),e\left(\frac{17}{20}\right))

First values

aa 1-111223344556677889910101111
χ1037(333,a) \chi_{ 1037 }(333, a) 1111e(1940)e\left(\frac{19}{40}\right)e(2380)e\left(\frac{23}{80}\right)e(1920)e\left(\frac{19}{20}\right)e(5180)e\left(\frac{51}{80}\right)e(6180)e\left(\frac{61}{80}\right)e(5780)e\left(\frac{57}{80}\right)e(1740)e\left(\frac{17}{40}\right)e(2340)e\left(\frac{23}{40}\right)e(980)e\left(\frac{9}{80}\right)e(116)e\left(\frac{1}{16}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ1037(333,a)   \chi_{ 1037 }(333,a) \; at   a=\;a = e.g. 2