Properties

Label 1037.516
Modulus 10371037
Conductor 10371037
Order 8080
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1037, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,68]))
 
pari: [g,chi] = znchar(Mod(516,1037))
 

Basic properties

Modulus: 10371037
Conductor: 10371037
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 8080
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
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) ...

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(ζ80)\Q(\zeta_{80})
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

(428,307)(428,307)(e(1516),e(1720))(e\left(\frac{15}{16}\right),e\left(\frac{17}{20}\right))

First values

aa 1-111223344556677889910101111
χ1037(516,a) \chi_{ 1037 }(516, a) 1111e(3940)e\left(\frac{39}{40}\right)e(380)e\left(\frac{3}{80}\right)e(1920)e\left(\frac{19}{20}\right)e(3180)e\left(\frac{31}{80}\right)e(180)e\left(\frac{1}{80}\right)e(7780)e\left(\frac{77}{80}\right)e(3740)e\left(\frac{37}{40}\right)e(340)e\left(\frac{3}{40}\right)e(2980)e\left(\frac{29}{80}\right)e(516)e\left(\frac{5}{16}\right)
sage: chi.jacobi_sum(n)
 
χ1037(516,a)   \chi_{ 1037 }(516,a) \; at   a=\;a = e.g. 2