Properties

Label 764.759
Modulus $764$
Conductor $764$
Order $38$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(764, base_ring=CyclotomicField(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([19,29]))
 
pari: [g,chi] = znchar(Mod(759,764))
 

Basic properties

Modulus: \(764\)
Conductor: \(764\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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 764.l

\(\chi_{764}(11,\cdot)\) \(\chi_{764}(31,\cdot)\) \(\chi_{764}(55,\cdot)\) \(\chi_{764}(139,\cdot)\) \(\chi_{764}(155,\cdot)\) \(\chi_{764}(159,\cdot)\) \(\chi_{764}(275,\cdot)\) \(\chi_{764}(419,\cdot)\) \(\chi_{764}(423,\cdot)\) \(\chi_{764}(543,\cdot)\) \(\chi_{764}(567,\cdot)\) \(\chi_{764}(587,\cdot)\) \(\chi_{764}(611,\cdot)\) \(\chi_{764}(639,\cdot)\) \(\chi_{764}(643,\cdot)\) \(\chi_{764}(695,\cdot)\) \(\chi_{764}(739,\cdot)\) \(\chi_{764}(759,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.38.687661045808093482376579225097085116287670624782479569073265850359469722789446885178751177457664.1

Values on generators

\((383,401)\) → \((-1,e\left(\frac{29}{38}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{3}{19}\right)\)\(1\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{1}{38}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 764 }(759,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{764}(759,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(759,r) e\left(\frac{r}{382}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 764 }(759,·),\chi_{ 764 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{764}(759,\cdot),\chi_{764}(1,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(759,r) \chi_{764}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 764 }(759,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{764}(759,·)) = \sum_{r \in \Z/764\Z} \chi_{764}(759,r) e\left(\frac{1 r + 2 r^{-1}}{764}\right) = 8.9575002692+-35.3723686246i \)