Properties

Label 764.407
Modulus $764$
Conductor $764$
Order $38$
Real no
Primitive yes
Minimal yes
Parity odd

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([19,20]))
 
pari: [g,chi] = znchar(Mod(407,764))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 764.k

\(\chi_{764}(107,\cdot)\) \(\chi_{764}(223,\cdot)\) \(\chi_{764}(227,\cdot)\) \(\chi_{764}(243,\cdot)\) \(\chi_{764}(327,\cdot)\) \(\chi_{764}(351,\cdot)\) \(\chi_{764}(371,\cdot)\) \(\chi_{764}(387,\cdot)\) \(\chi_{764}(407,\cdot)\) \(\chi_{764}(451,\cdot)\) \(\chi_{764}(503,\cdot)\) \(\chi_{764}(507,\cdot)\) \(\chi_{764}(535,\cdot)\) \(\chi_{764}(559,\cdot)\) \(\chi_{764}(579,\cdot)\) \(\chi_{764}(603,\cdot)\) \(\chi_{764}(723,\cdot)\) \(\chi_{764}(727,\cdot)\)

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: 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1

Values on generators

\((383,401)\) → \((-1,e\left(\frac{10}{19}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{6}{19}\right)\)\(-1\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{1}{19}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 764 }(407,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{764}(407,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(407,r) e\left(\frac{r}{382}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 764 }(407,·),\chi_{ 764 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{764}(407,\cdot),\chi_{764}(1,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(407,r) \chi_{764}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 764 }(407,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{764}(407,·)) = \sum_{r \in \Z/764\Z} \chi_{764}(407,r) e\left(\frac{1 r + 2 r^{-1}}{764}\right) = -5.7295739295+10.5873253017i \)