Properties

Label 953.22
Modulus $953$
Conductor $953$
Order $952$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(953, base_ring=CyclotomicField(952))
 
M = H._module
 
chi = DirichletCharacter(H, M([437]))
 
pari: [g,chi] = znchar(Mod(22,953))
 

Basic properties

Modulus: \(953\)
Conductor: \(953\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(952\)
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 953.p

\(\chi_{953}(3,\cdot)\) \(\chi_{953}(5,\cdot)\) \(\chi_{953}(6,\cdot)\) \(\chi_{953}(10,\cdot)\) \(\chi_{953}(11,\cdot)\) \(\chi_{953}(12,\cdot)\) \(\chi_{953}(19,\cdot)\) \(\chi_{953}(20,\cdot)\) \(\chi_{953}(22,\cdot)\) \(\chi_{953}(23,\cdot)\) \(\chi_{953}(24,\cdot)\) \(\chi_{953}(27,\cdot)\) \(\chi_{953}(35,\cdot)\) \(\chi_{953}(38,\cdot)\) \(\chi_{953}(43,\cdot)\) \(\chi_{953}(44,\cdot)\) \(\chi_{953}(46,\cdot)\) \(\chi_{953}(47,\cdot)\) \(\chi_{953}(48,\cdot)\) \(\chi_{953}(51,\cdot)\) \(\chi_{953}(54,\cdot)\) \(\chi_{953}(61,\cdot)\) \(\chi_{953}(65,\cdot)\) \(\chi_{953}(70,\cdot)\) \(\chi_{953}(75,\cdot)\) \(\chi_{953}(76,\cdot)\) \(\chi_{953}(77,\cdot)\) \(\chi_{953}(80,\cdot)\) \(\chi_{953}(85,\cdot)\) \(\chi_{953}(87,\cdot)\) ...

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: $\Q(\zeta_{952})$
Fixed field: Number field defined by a degree 952 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{437}{952}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 953 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{57}{68}\right)\)\(e\left(\frac{437}{952}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{547}{952}\right)\)\(e\left(\frac{283}{952}\right)\)\(e\left(\frac{15}{238}\right)\)\(e\left(\frac{35}{68}\right)\)\(e\left(\frac{437}{476}\right)\)\(e\left(\frac{393}{952}\right)\)\(e\left(\frac{723}{952}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 953 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 953 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 953 }(22,·),\chi_{ 953 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 953 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2