Properties

Label 953.2
Modulus $953$
Conductor $953$
Order $68$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(953, base_ring=CyclotomicField(68))
 
M = H._module
 
chi = DirichletCharacter(H, M([9]))
 
pari: [g,chi] = znchar(Mod(2,953))
 

Basic properties

Modulus: \(953\)
Conductor: \(953\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
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 953.k

\(\chi_{953}(2,\cdot)\) \(\chi_{953}(8,\cdot)\) \(\chi_{953}(32,\cdot)\) \(\chi_{953}(67,\cdot)\) \(\chi_{953}(119,\cdot)\) \(\chi_{953}(128,\cdot)\) \(\chi_{953}(138,\cdot)\) \(\chi_{953}(142,\cdot)\) \(\chi_{953}(255,\cdot)\) \(\chi_{953}(268,\cdot)\) \(\chi_{953}(302,\cdot)\) \(\chi_{953}(366,\cdot)\) \(\chi_{953}(385,\cdot)\) \(\chi_{953}(401,\cdot)\) \(\chi_{953}(441,\cdot)\) \(\chi_{953}(476,\cdot)\) \(\chi_{953}(477,\cdot)\) \(\chi_{953}(512,\cdot)\) \(\chi_{953}(552,\cdot)\) \(\chi_{953}(568,\cdot)\) \(\chi_{953}(587,\cdot)\) \(\chi_{953}(651,\cdot)\) \(\chi_{953}(685,\cdot)\) \(\chi_{953}(698,\cdot)\) \(\chi_{953}(811,\cdot)\) \(\chi_{953}(815,\cdot)\) \(\chi_{953}(825,\cdot)\) \(\chi_{953}(834,\cdot)\) \(\chi_{953}(886,\cdot)\) \(\chi_{953}(921,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\(3\) → \(e\left(\frac{9}{68}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 953 }(2, a) \) \(1\)\(1\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{9}{68}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{55}{68}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{61}{68}\right)\)\(e\left(\frac{11}{68}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 953 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

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

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 953 }(2,·),\chi_{ 953 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

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