Properties

Label 82.15
Modulus $82$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(82, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([37]))
 
pari: [g,chi] = znchar(Mod(15,82))
 

Basic properties

Modulus: \(82\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 82.h

\(\chi_{82}(7,\cdot)\) \(\chi_{82}(11,\cdot)\) \(\chi_{82}(13,\cdot)\) \(\chi_{82}(15,\cdot)\) \(\chi_{82}(17,\cdot)\) \(\chi_{82}(19,\cdot)\) \(\chi_{82}(29,\cdot)\) \(\chi_{82}(35,\cdot)\) \(\chi_{82}(47,\cdot)\) \(\chi_{82}(53,\cdot)\) \(\chi_{82}(63,\cdot)\) \(\chi_{82}(65,\cdot)\) \(\chi_{82}(67,\cdot)\) \(\chi_{82}(69,\cdot)\) \(\chi_{82}(71,\cdot)\) \(\chi_{82}(75,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\(47\) → \(e\left(\frac{37}{40}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 82 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{40}\right)\)\(-i\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{19}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 82 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 82 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 82 }(15,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 82 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2