Properties

Label 961.838
Modulus $961$
Conductor $961$
Order $31$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(62))
 
M = H._module
 
chi = DirichletCharacter(H, M([36]))
 
pari: [g,chi] = znchar(Mod(838,961))
 

Basic properties

Modulus: \(961\)
Conductor: \(961\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(31\)
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 961.i

\(\chi_{961}(32,\cdot)\) \(\chi_{961}(63,\cdot)\) \(\chi_{961}(94,\cdot)\) \(\chi_{961}(125,\cdot)\) \(\chi_{961}(156,\cdot)\) \(\chi_{961}(187,\cdot)\) \(\chi_{961}(218,\cdot)\) \(\chi_{961}(249,\cdot)\) \(\chi_{961}(280,\cdot)\) \(\chi_{961}(311,\cdot)\) \(\chi_{961}(342,\cdot)\) \(\chi_{961}(373,\cdot)\) \(\chi_{961}(404,\cdot)\) \(\chi_{961}(435,\cdot)\) \(\chi_{961}(466,\cdot)\) \(\chi_{961}(497,\cdot)\) \(\chi_{961}(528,\cdot)\) \(\chi_{961}(559,\cdot)\) \(\chi_{961}(590,\cdot)\) \(\chi_{961}(621,\cdot)\) \(\chi_{961}(652,\cdot)\) \(\chi_{961}(683,\cdot)\) \(\chi_{961}(714,\cdot)\) \(\chi_{961}(745,\cdot)\) \(\chi_{961}(776,\cdot)\) \(\chi_{961}(807,\cdot)\) \(\chi_{961}(838,\cdot)\) \(\chi_{961}(869,\cdot)\) \(\chi_{961}(900,\cdot)\) \(\chi_{961}(931,\cdot)\)

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

Related number fields

Field of values: $\Q(\zeta_{31})$
Fixed field: 31.31.303180754947920226773910663600827932461415650100367091342054488471568688197420529491484801.1

Values on generators

\(3\) → \(e\left(\frac{18}{31}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 961 }(838, a) \) \(1\)\(1\)\(e\left(\frac{10}{31}\right)\)\(e\left(\frac{18}{31}\right)\)\(e\left(\frac{20}{31}\right)\)\(e\left(\frac{15}{31}\right)\)\(e\left(\frac{28}{31}\right)\)\(e\left(\frac{2}{31}\right)\)\(e\left(\frac{30}{31}\right)\)\(e\left(\frac{5}{31}\right)\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{17}{31}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 961 }(838,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 961 }(838,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 961 }(838,·),\chi_{ 961 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 961 }(838,·)) \;\) at \(\; a,b = \) e.g. 1,2