Properties

Label 1682.57
Modulus $1682$
Conductor $841$
Order $58$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([25]))
 
pari: [g,chi] = znchar(Mod(57,1682))
 

Basic properties

Modulus: \(1682\)
Conductor: \(841\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{841}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1682.h

\(\chi_{1682}(57,\cdot)\) \(\chi_{1682}(115,\cdot)\) \(\chi_{1682}(173,\cdot)\) \(\chi_{1682}(231,\cdot)\) \(\chi_{1682}(289,\cdot)\) \(\chi_{1682}(347,\cdot)\) \(\chi_{1682}(405,\cdot)\) \(\chi_{1682}(463,\cdot)\) \(\chi_{1682}(521,\cdot)\) \(\chi_{1682}(579,\cdot)\) \(\chi_{1682}(637,\cdot)\) \(\chi_{1682}(695,\cdot)\) \(\chi_{1682}(753,\cdot)\) \(\chi_{1682}(811,\cdot)\) \(\chi_{1682}(869,\cdot)\) \(\chi_{1682}(927,\cdot)\) \(\chi_{1682}(985,\cdot)\) \(\chi_{1682}(1043,\cdot)\) \(\chi_{1682}(1101,\cdot)\) \(\chi_{1682}(1159,\cdot)\) \(\chi_{1682}(1217,\cdot)\) \(\chi_{1682}(1275,\cdot)\) \(\chi_{1682}(1333,\cdot)\) \(\chi_{1682}(1391,\cdot)\) \(\chi_{1682}(1449,\cdot)\) \(\chi_{1682}(1507,\cdot)\) \(\chi_{1682}(1565,\cdot)\) \(\chi_{1682}(1623,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\(843\) → \(e\left(\frac{25}{58}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1682 }(57, a) \) \(1\)\(1\)\(e\left(\frac{27}{58}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{53}{58}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{37}{58}\right)\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{23}{58}\right)\)\(e\left(\frac{31}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1682 }(57,a) \;\) at \(\;a = \) e.g. 2