Properties

Label 87.55
Modulus $87$
Conductor $29$
Order $28$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,19]))
 
pari: [g,chi] = znchar(Mod(55,87))
 

Basic properties

Modulus: \(87\)
Conductor: \(29\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{29}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 87.l

\(\chi_{87}(10,\cdot)\) \(\chi_{87}(19,\cdot)\) \(\chi_{87}(31,\cdot)\) \(\chi_{87}(37,\cdot)\) \(\chi_{87}(40,\cdot)\) \(\chi_{87}(43,\cdot)\) \(\chi_{87}(55,\cdot)\) \(\chi_{87}(61,\cdot)\) \(\chi_{87}(73,\cdot)\) \(\chi_{87}(76,\cdot)\) \(\chi_{87}(79,\cdot)\) \(\chi_{87}(85,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((59,31)\) → \((1,e\left(\frac{19}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 87 }(55, a) \) \(-1\)\(1\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{5}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 87 }(55,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 87 }(55,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 87 }(55,·),\chi_{ 87 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 87 }(55,·)) \;\) at \(\; a,b = \) e.g. 1,2