Properties

Label 740.7
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([18,9,32]))
 
pari: [g,chi] = znchar(Mod(7,740))
 

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.cf

\(\chi_{740}(7,\cdot)\) \(\chi_{740}(83,\cdot)\) \(\chi_{740}(107,\cdot)\) \(\chi_{740}(123,\cdot)\) \(\chi_{740}(127,\cdot)\) \(\chi_{740}(303,\cdot)\) \(\chi_{740}(367,\cdot)\) \(\chi_{740}(403,\cdot)\) \(\chi_{740}(423,\cdot)\) \(\chi_{740}(527,\cdot)\) \(\chi_{740}(567,\cdot)\) \(\chi_{740}(663,\cdot)\)

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: Number field defined by a degree 36 polynomial

Values on generators

\((371,297,261)\) → \((-1,i,e\left(\frac{8}{9}\right))\)

First values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 740 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 740 }(7,·),\chi_{ 740 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 740 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2