Properties

Label 764.341
Modulus $764$
Conductor $191$
Order $19$
Real no
Primitive no
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([0,14]))
 
pari: [g,chi] = znchar(Mod(341,764))
 

Basic properties

Modulus: \(764\)
Conductor: \(191\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{191}(150,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 764.i

\(\chi_{764}(5,\cdot)\) \(\chi_{764}(25,\cdot)\) \(\chi_{764}(69,\cdot)\) \(\chi_{764}(121,\cdot)\) \(\chi_{764}(125,\cdot)\) \(\chi_{764}(153,\cdot)\) \(\chi_{764}(177,\cdot)\) \(\chi_{764}(197,\cdot)\) \(\chi_{764}(221,\cdot)\) \(\chi_{764}(341,\cdot)\) \(\chi_{764}(345,\cdot)\) \(\chi_{764}(489,\cdot)\) \(\chi_{764}(605,\cdot)\) \(\chi_{764}(609,\cdot)\) \(\chi_{764}(625,\cdot)\) \(\chi_{764}(709,\cdot)\) \(\chi_{764}(733,\cdot)\) \(\chi_{764}(753,\cdot)\)

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: 19.19.114445997944945591651333831028437092270721.1

Values on generators

\((383,401)\) → \((1,e\left(\frac{7}{19}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(1\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{14}{19}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 764 }(341,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{764}(341,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(341,r) e\left(\frac{r}{382}\right) = 25.4318079485+10.8269637698i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 764 }(341,·),\chi_{ 764 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{764}(341,\cdot),\chi_{764}(1,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(341,r) \chi_{764}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 764 }(341,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{764}(341,·)) = \sum_{r \in \Z/764\Z} \chi_{764}(341,r) e\left(\frac{1 r + 2 r^{-1}}{764}\right) = 0.0 \)