Properties

Label 760.637
Modulus $760$
Conductor $760$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(760, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,18,9,34]))
 
pari: [g,chi] = znchar(Mod(637,760))
 

Basic properties

Modulus: \(760\)
Conductor: \(760\)
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 760.cs

\(\chi_{760}(13,\cdot)\) \(\chi_{760}(53,\cdot)\) \(\chi_{760}(117,\cdot)\) \(\chi_{760}(173,\cdot)\) \(\chi_{760}(317,\cdot)\) \(\chi_{760}(333,\cdot)\) \(\chi_{760}(357,\cdot)\) \(\chi_{760}(413,\cdot)\) \(\chi_{760}(477,\cdot)\) \(\chi_{760}(573,\cdot)\) \(\chi_{760}(637,\cdot)\) \(\chi_{760}(717,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1

Values on generators

\((191,381,457,401)\) → \((1,-1,i,e\left(\frac{17}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 760 }(637,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 760 }(637,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 760 }(637,·),\chi_{ 760 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 760 }(637,·)) \;\) at \(\; a,b = \) e.g. 1,2