Properties

Modulus 389
Conductor 389
Order 97
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 389.d

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(389)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([76]))
 
pari: [g,chi] = znchar(Mod(78,389))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 389
Conductor = 389
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 97
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 389.d
Orbit index = 4

Galois orbit

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

\(\chi_{389}(5,\cdot)\) \(\chi_{389}(6,\cdot)\) \(\chi_{389}(7,\cdot)\) \(\chi_{389}(11,\cdot)\) \(\chi_{389}(13,\cdot)\) \(\chi_{389}(16,\cdot)\) \(\chi_{389}(17,\cdot)\) \(\chi_{389}(25,\cdot)\) \(\chi_{389}(30,\cdot)\) \(\chi_{389}(35,\cdot)\) \(\chi_{389}(36,\cdot)\) \(\chi_{389}(42,\cdot)\) \(\chi_{389}(49,\cdot)\) \(\chi_{389}(55,\cdot)\) \(\chi_{389}(58,\cdot)\) \(\chi_{389}(65,\cdot)\) \(\chi_{389}(66,\cdot)\) \(\chi_{389}(67,\cdot)\) \(\chi_{389}(69,\cdot)\) \(\chi_{389}(73,\cdot)\) \(\chi_{389}(74,\cdot)\) \(\chi_{389}(76,\cdot)\) \(\chi_{389}(77,\cdot)\) \(\chi_{389}(78,\cdot)\) \(\chi_{389}(79,\cdot)\) \(\chi_{389}(80,\cdot)\) \(\chi_{389}(81,\cdot)\) \(\chi_{389}(85,\cdot)\) \(\chi_{389}(91,\cdot)\) \(\chi_{389}(93,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{76}{97}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{76}{97}\right)\)\(e\left(\frac{32}{97}\right)\)\(e\left(\frac{55}{97}\right)\)\(e\left(\frac{79}{97}\right)\)\(e\left(\frac{11}{97}\right)\)\(e\left(\frac{96}{97}\right)\)\(e\left(\frac{34}{97}\right)\)\(e\left(\frac{64}{97}\right)\)\(e\left(\frac{58}{97}\right)\)\(e\left(\frac{74}{97}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{97})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 389 }(78,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{389}(78,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(78,r) e\left(\frac{2r}{389}\right) = 17.1965656858+-9.6580602925i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 389 }(78,·),\chi_{ 389 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{389}(78,\cdot),\chi_{389}(1,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(78,r) \chi_{389}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 389 }(78,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{389}(78,·)) = \sum_{r \in \Z/389\Z} \chi_{389}(78,r) e\left(\frac{1 r + 2 r^{-1}}{389}\right) = -16.6959704962+13.5052146274i \)