Properties

Label 953.7
Modulus $953$
Conductor $953$
Order $238$
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(238))
 
M = H._module
 
chi = DirichletCharacter(H, M([201]))
 
pari: [g,chi] = znchar(Mod(7,953))
 

Basic properties

Modulus: \(953\)
Conductor: \(953\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(238\)
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.n

\(\chi_{953}(7,\cdot)\) \(\chi_{953}(26,\cdot)\) \(\chi_{953}(29,\cdot)\) \(\chi_{953}(34,\cdot)\) \(\chi_{953}(50,\cdot)\) \(\chi_{953}(60,\cdot)\) \(\chi_{953}(72,\cdot)\) \(\chi_{953}(81,\cdot)\) \(\chi_{953}(82,\cdot)\) \(\chi_{953}(105,\cdot)\) \(\chi_{953}(110,\cdot)\) \(\chi_{953}(112,\cdot)\) \(\chi_{953}(118,\cdot)\) \(\chi_{953}(126,\cdot)\) \(\chi_{953}(127,\cdot)\) \(\chi_{953}(129,\cdot)\) \(\chi_{953}(132,\cdot)\) \(\chi_{953}(157,\cdot)\) \(\chi_{953}(158,\cdot)\) \(\chi_{953}(166,\cdot)\) \(\chi_{953}(169,\cdot)\) \(\chi_{953}(196,\cdot)\) \(\chi_{953}(206,\cdot)\) \(\chi_{953}(212,\cdot)\) \(\chi_{953}(218,\cdot)\) \(\chi_{953}(231,\cdot)\) \(\chi_{953}(262,\cdot)\) \(\chi_{953}(277,\cdot)\) \(\chi_{953}(289,\cdot)\) \(\chi_{953}(321,\cdot)\) ...

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: $\Q(\zeta_{119})$
Fixed field: Number field defined by a degree 238 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{201}{238}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 953 }(7, a) \) \(1\)\(1\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{201}{238}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{165}{238}\right)\)\(e\left(\frac{61}{238}\right)\)\(e\left(\frac{1}{119}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{82}{119}\right)\)\(e\left(\frac{25}{238}\right)\)\(e\left(\frac{155}{238}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 953 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 953 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 953 }(7,·),\chi_{ 953 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 953 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2