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([28]))
 
pari: [g,chi] = znchar(Mod(77,389))
 

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

\(2\) → \(e\left(\frac{28}{97}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{28}{97}\right)\)\(e\left(\frac{22}{97}\right)\)\(e\left(\frac{56}{97}\right)\)\(e\left(\frac{24}{97}\right)\)\(e\left(\frac{50}{97}\right)\)\(e\left(\frac{66}{97}\right)\)\(e\left(\frac{84}{97}\right)\)\(e\left(\frac{44}{97}\right)\)\(e\left(\frac{52}{97}\right)\)\(e\left(\frac{63}{97}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 389 }(77,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{389}(77,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(77,r) e\left(\frac{2r}{389}\right) = -18.0099540965+-8.0399971046i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 389 }(77,·),\chi_{ 389 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{389}(77,\cdot),\chi_{389}(1,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(77,r) \chi_{389}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 389 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{389}(77,·)) = \sum_{r \in \Z/389\Z} \chi_{389}(77,r) e\left(\frac{1 r + 2 r^{-1}}{389}\right) = 12.9501479041+16.5508661935i \)