Properties

Label 740.39
Modulus $740$
Conductor $740$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,18,1]))
 
pari: [g,chi] = znchar(Mod(39,740))
 

Basic properties

Modulus: \(740\)
Conductor: \(740\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 740.ca

\(\chi_{740}(19,\cdot)\) \(\chi_{740}(39,\cdot)\) \(\chi_{740}(59,\cdot)\) \(\chi_{740}(79,\cdot)\) \(\chi_{740}(239,\cdot)\) \(\chi_{740}(279,\cdot)\) \(\chi_{740}(439,\cdot)\) \(\chi_{740}(459,\cdot)\) \(\chi_{740}(479,\cdot)\) \(\chi_{740}(499,\cdot)\) \(\chi_{740}(579,\cdot)\) \(\chi_{740}(679,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.2021157996966699236888348986167488408933250634681517576814592000000000000000000.1

Values on generators

\((371,297,261)\) → \((-1,-1,e\left(\frac{1}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 740 }(39, a) \) \(1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 740 }(39,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 740 }(39,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 740 }(39,·),\chi_{ 740 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 740 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2