Properties

Label 74.25
Modulus $74$
Conductor $37$
Order $18$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(74, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5]))
 
pari: [g,chi] = znchar(Mod(25,74))
 

Basic properties

Modulus: \(74\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 74.h

\(\chi_{74}(3,\cdot)\) \(\chi_{74}(21,\cdot)\) \(\chi_{74}(25,\cdot)\) \(\chi_{74}(41,\cdot)\) \(\chi_{74}(65,\cdot)\) \(\chi_{74}(67,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{37})^+\)

Values on generators

\(39\) → \(e\left(\frac{5}{18}\right)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 74 }(25, a) \) \(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 74 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 74 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 74 }(25,·),\chi_{ 74 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 74 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2