Properties

Label 1037.106
Modulus 10371037
Conductor 10371037
Order 6060
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1037, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,34]))
 
pari: [g,chi] = znchar(Mod(106,1037))
 

Basic properties

Modulus: 10371037
Conductor: 10371037
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 6060
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.ch

χ1037(4,)\chi_{1037}(4,\cdot) χ1037(106,)\chi_{1037}(106,\cdot) χ1037(293,)\chi_{1037}(293,\cdot) χ1037(310,)\chi_{1037}(310,\cdot) χ1037(344,)\chi_{1037}(344,\cdot) χ1037(370,)\chi_{1037}(370,\cdot) χ1037(412,)\chi_{1037}(412,\cdot) χ1037(446,)\chi_{1037}(446,\cdot) χ1037(463,)\chi_{1037}(463,\cdot) χ1037(472,)\chi_{1037}(472,\cdot) χ1037(659,)\chi_{1037}(659,\cdot) χ1037(676,)\chi_{1037}(676,\cdot) χ1037(710,)\chi_{1037}(710,\cdot) χ1037(778,)\chi_{1037}(778,\cdot) χ1037(812,)\chi_{1037}(812,\cdot) χ1037(829,)\chi_{1037}(829,\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(ζ60)\Q(\zeta_{60})
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

(428,307)(428,307)(i,e(1730))(-i,e\left(\frac{17}{30}\right))

First values

aa 1-111223344556677889910101111
χ1037(106,a) \chi_{ 1037 }(106, a) 1111e(115)e\left(\frac{1}{15}\right)e(320)e\left(\frac{3}{20}\right)e(215)e\left(\frac{2}{15}\right)e(1360)e\left(\frac{13}{60}\right)e(1360)e\left(\frac{13}{60}\right)e(160)e\left(\frac{1}{60}\right)e(15)e\left(\frac{1}{5}\right)e(310)e\left(\frac{3}{10}\right)e(1760)e\left(\frac{17}{60}\right)i-i
sage: chi.jacobi_sum(n)
 
χ1037(106,a)   \chi_{ 1037 }(106,a) \; at   a=\;a = e.g. 2