Properties

Label 1856.137
Modulus $1856$
Conductor $928$
Order $56$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,34]))
 
pari: [g,chi] = znchar(Mod(137,1856))
 

Basic properties

Modulus: \(1856\)
Conductor: \(928\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{928}(253,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1856.cg

\(\chi_{1856}(137,\cdot)\) \(\chi_{1856}(153,\cdot)\) \(\chi_{1856}(345,\cdot)\) \(\chi_{1856}(425,\cdot)\) \(\chi_{1856}(537,\cdot)\) \(\chi_{1856}(553,\cdot)\) \(\chi_{1856}(569,\cdot)\) \(\chi_{1856}(649,\cdot)\) \(\chi_{1856}(665,\cdot)\) \(\chi_{1856}(681,\cdot)\) \(\chi_{1856}(793,\cdot)\) \(\chi_{1856}(873,\cdot)\) \(\chi_{1856}(1065,\cdot)\) \(\chi_{1856}(1081,\cdot)\) \(\chi_{1856}(1273,\cdot)\) \(\chi_{1856}(1353,\cdot)\) \(\chi_{1856}(1465,\cdot)\) \(\chi_{1856}(1481,\cdot)\) \(\chi_{1856}(1497,\cdot)\) \(\chi_{1856}(1577,\cdot)\) \(\chi_{1856}(1593,\cdot)\) \(\chi_{1856}(1609,\cdot)\) \(\chi_{1856}(1721,\cdot)\) \(\chi_{1856}(1801,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((639,581,321)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{17}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1856 }(137, a) \) \(-1\)\(1\)\(e\left(\frac{9}{56}\right)\)\(e\left(\frac{41}{56}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{31}{56}\right)\)\(e\left(\frac{25}{28}\right)\)\(i\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{11}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1856 }(137,a) \;\) at \(\;a = \) e.g. 2