Properties

Label 2151.283
Modulus $2151$
Conductor $2151$
Order $21$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,24]))
 
pari: [g,chi] = znchar(Mod(283,2151))
 

Basic properties

Modulus: \(2151\)
Conductor: \(2151\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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 2151.n

\(\chi_{2151}(283,\cdot)\) \(\chi_{2151}(337,\cdot)\) \(\chi_{2151}(502,\cdot)\) \(\chi_{2151}(679,\cdot)\) \(\chi_{2151}(727,\cdot)\) \(\chi_{2151}(817,\cdot)\) \(\chi_{2151}(1219,\cdot)\) \(\chi_{2151}(1444,\cdot)\) \(\chi_{2151}(1534,\cdot)\) \(\chi_{2151}(1717,\cdot)\) \(\chi_{2151}(1771,\cdot)\) \(\chi_{2151}(2113,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((479,1441)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2151 }(283, a) \) \(1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2151 }(283,a) \;\) at \(\;a = \) e.g. 2