Properties

Label 1037.852
Modulus $1037$
Conductor $1037$
Order $120$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1037, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,62]))
 
pari: [g,chi] = znchar(Mod(852,1037))
 

Basic properties

Modulus: \(1037\)
Conductor: \(1037\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
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 1037.cm

\(\chi_{1037}(26,\cdot)\) \(\chi_{1037}(104,\cdot)\) \(\chi_{1037}(189,\cdot)\) \(\chi_{1037}(213,\cdot)\) \(\chi_{1037}(246,\cdot)\) \(\chi_{1037}(287,\cdot)\) \(\chi_{1037}(331,\cdot)\) \(\chi_{1037}(349,\cdot)\) \(\chi_{1037}(359,\cdot)\) \(\chi_{1037}(372,\cdot)\) \(\chi_{1037}(376,\cdot)\) \(\chi_{1037}(417,\cdot)\) \(\chi_{1037}(434,\cdot)\) \(\chi_{1037}(444,\cdot)\) \(\chi_{1037}(518,\cdot)\) \(\chi_{1037}(542,\cdot)\) \(\chi_{1037}(559,\cdot)\) \(\chi_{1037}(580,\cdot)\) \(\chi_{1037}(604,\cdot)\) \(\chi_{1037}(627,\cdot)\) \(\chi_{1037}(654,\cdot)\) \(\chi_{1037}(689,\cdot)\) \(\chi_{1037}(722,\cdot)\) \(\chi_{1037}(739,\cdot)\) \(\chi_{1037}(763,\cdot)\) \(\chi_{1037}(767,\cdot)\) \(\chi_{1037}(852,\cdot)\) \(\chi_{1037}(909,\cdot)\) \(\chi_{1037}(950,\cdot)\) \(\chi_{1037}(978,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((428,307)\) → \((e\left(\frac{7}{8}\right),e\left(\frac{31}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1037 }(852, a) \) \(-1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{89}{120}\right)\)\(e\left(\frac{89}{120}\right)\)\(e\left(\frac{113}{120}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{61}{120}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1037 }(852,a) \;\) at \(\;a = \) e.g. 2