Properties

Label 637.15
Modulus $637$
Conductor $637$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([60,7]))
 
pari: [g,chi] = znchar(Mod(15,637))
 

Basic properties

Modulus: \(637\)
Conductor: \(637\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 637.cf

\(\chi_{637}(15,\cdot)\) \(\chi_{637}(71,\cdot)\) \(\chi_{637}(85,\cdot)\) \(\chi_{637}(106,\cdot)\) \(\chi_{637}(141,\cdot)\) \(\chi_{637}(162,\cdot)\) \(\chi_{637}(176,\cdot)\) \(\chi_{637}(232,\cdot)\) \(\chi_{637}(253,\cdot)\) \(\chi_{637}(267,\cdot)\) \(\chi_{637}(288,\cdot)\) \(\chi_{637}(323,\cdot)\) \(\chi_{637}(358,\cdot)\) \(\chi_{637}(379,\cdot)\) \(\chi_{637}(414,\cdot)\) \(\chi_{637}(435,\cdot)\) \(\chi_{637}(449,\cdot)\) \(\chi_{637}(470,\cdot)\) \(\chi_{637}(505,\cdot)\) \(\chi_{637}(526,\cdot)\) \(\chi_{637}(561,\cdot)\) \(\chi_{637}(596,\cdot)\) \(\chi_{637}(617,\cdot)\) \(\chi_{637}(631,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((248,197)\) → \((e\left(\frac{5}{7}\right),e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 637 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{5}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 637 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 637 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 637 }(15,·),\chi_{ 637 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 637 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2